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Guide6 Aug 2026Linear Algebra Algorithms over Fields and RingsExact Gaussian elimination, fraction-free elimination, determinant and characteristic polynomial algorithms, and kernel and image computation over fields and over ℤ.5 min readRead MoreGuide6 Aug 2026Polynomial Arithmetic and GCD in Unique Factorisation DomainsDense and sparse polynomial representation, multiplication algorithms, pseudo-division over a UFD, primitive parts and content, and the subresultant and modular remedies for coefficient growth.4 min readRead MoreGuide6 Aug 2026Algebraic Numbers and Number FieldsAlgebraic numbers and integers, minimal polynomials, number fields as finite extensions of Q, real and complex embeddings, the signature, and the primitive element theorem.5 min readRead MoreGuide6 Aug 2026Quadratic Fields and Binary Quadratic FormsQuadratic fields, their discriminants and integral bases, prime decomposition by the Kronecker symbol, binary quadratic forms, reduction, and the correspondence between form classes and ideal classes.5 min readRead MoreGuide6 Aug 2026Computing the Maximal Order: the Round 2 AlgorithmThe Pohst-Zassenhaus theorem, the Dedekind criterion, the radical and the ring of multipliers, and the Round 2 algorithm for computing the ring of integers.5 min readRead MoreGuide6 Aug 2026Elliptic Curves: Definitions and the Group LawWeierstrass forms, the discriminant and j-invariant, the group law with explicit formulas, torsion, and the structure of the group of points over finite fields.5 min readRead MoreGuide6 Aug 2026Primality Testing versus FactoringThe separation between primality testing and factoring, compositeness tests versus primality proofs, probable primes, pseudoprimes, and how to select a testing strategy.5 min readRead MoreGuide6 Aug 2026Classical Factoring: Trial Division, Fermat and LehmanTrial division and wheel factorisation, Fermat's difference of squares method, Lehman's improvement, and the role of these methods as a preprocessing stage.5 min readRead MoreGuide6 Aug 2026Homological Algebra: Discipline OverviewStructural overview of homological algebra: the failure of exactness, resolutions and derived functors, the central invariants Ext and Tor, and how the machinery specialises to groups, Lie algebras and topology.5 min readRead MoreGuide6 Aug 2026Modules and Module HomomorphismsModules over a ring, submodules and quotients, module homomorphisms, exact sequences, the short five lemma and the snake lemma.4 min readRead MoreGuide6 Aug 2026Categories and MorphismsCategories, objects and morphisms, monomorphisms and epimorphisms defined by cancellation, isomorphisms, and why the arrow-theoretic definitions differ from the element-based ones.3 min readRead MoreGuide6 Aug 2026Extensions of Modules and the Baer SumExtensions of modules, equivalence of extensions, the Baer sum defined by pullback and pushout, and the resulting abelian group structure on Ext.4 min readRead MoreGuide6 Aug 2026Chain Complexes and HomologyChain and cochain complexes, cycles and boundaries, homology as a functor, and the abelian category of complexes.3 min readRead MoreGuide6 Aug 2026Double Complexes and Total ComplexesDouble complexes, the sign convention, the total complex by sum or product, and the two filtrations that give rise to spectral sequences.3 min readRead MoreGuide6 Aug 2026The Group Ring and the Augmentation IdealThe group ring, modules over it as representations, the augmentation map and its kernel, and the relation between the augmentation ideal and the abelianisation.3 min readRead MoreGuide6 Aug 2026Lie Algebras and the Universal Enveloping AlgebraLie algebras, representations as modules, the universal enveloping algebra and its universal property, the Poincare-Birkhoff-Witt theorem, and the augmentation ideal.4 min readRead MoreGuide6 Aug 2026Exact Couples and Spectral SequencesSpectral sequences as successive approximations, pages and differentials, exact couples and their derivation, and the standard sources of spectral sequences.4 min readRead MoreGuide6 Aug 2026Projective Classes of EpimorphismsProjective classes, relative projectives, allowable epimorphisms, relative exactness, and the examples that motivate relative homological algebra.4 min readRead MoreGuide6 Aug 2026Homological Algebra and Algebraic TopologyThe topological origins of homological algebra, singular and cellular chains, the Eilenberg-Zilber and Kunneth theorems in topology, classifying spaces, and the dictionary between the two subjects.4 min readRead MoreGuide6 Aug 2026Computation and Authoritative SourcesSoftware for computing resolutions, Ext, Tor and group cohomology, together with the authoritative databases and the KEVOS policy on reproducing computed data.4 min readRead MoreGuide6 Aug 2026Universal Algebra: Discipline OverviewStructural overview of universal algebra: what an algebra of arbitrary type is, why congruences rather than substructures carry the quotient theory, how Birkhoff's two theorems organise the subject, and how the five streams of this collection fit together.8 min readRead MoreGuide6 Aug 2026Posets and the Two Definitions of a LatticeThe order-theoretic and algebraic definitions of a lattice, the equivalence between them, and why universal algebra needs the algebraic form: only an equationally defined class is a variety.5 min readRead MoreGuide6 Aug 2026Algebras, Types and SignaturesAlgebras of arbitrary type: signatures, arities, the significance of nullary operations, and how the choice of type determines subalgebras, homomorphisms and the whole subsequent theory.4 min readRead MoreGuide6 Aug 2026Terms, Term Algebras and Term OperationsTerms as syntactic objects, the term algebra T(X) and its absolute freeness, the distinction between a term and the operation it induces, and why the term algebra is the reference object for equational reasoning.6 min readRead MoreGuide6 Aug 2026Steiner Triple Systems, Squags and SloopsSteiner triple systems recast as algebras: squags (Steiner quasigroups) and sloops (Steiner loops), the equational axioms for each, and what the variety structure delivers that the combinatorial description does not.6 min readRead MoreGuide6 Aug 2026Boolean Algebras: Axioms and StructureBoolean algebras as an equational class, the two-element algebra as the unique subdirectly irreducible member, atoms and atomlessness, and why the variety is arithmetical.6 min readRead MoreGuide6 Aug 2026Boolean PowersBoolean powers A[B]*, their construction as locally constant functions on a Stone space, the identities they preserve, and filtered Boolean powers as the refinement that carries the decidability results.6 min readRead MoreGuide6 Aug 2026First-order Languages and StructuresFirst-order languages with relation and operation symbols, structures as the semantic objects, the syntax of terms and formulas, free and bound variables, and how algebras sit inside the wider class of structures.5 min readRead MoreGuide6 Aug 2026Recent Developments: the 1981 FrontierThe source's Recent Developments chapter: the commutator programme, the classification of varieties by congruence conditions, decidability questions, Boolean constructions, structure theory, and the applications to computer science and model theory as they stood in 1981.5 min readRead MoreArticle7 Aug 2026What Universal Algebra Is: Scope and MethodUniversal algebra studies what all algebraic structures have in common by stripping away the particular operations of groups, rings and lattices and asking which theorems survive. This page sets out the scope of the subject, the method it uses, and the shape of the results it produces.4 min readRead MoreArticle7 Aug 2026Lattices as Algebras: the Equational DefinitionA lattice can be defined purely equationally, as a set with two binary operations satisfying four pairs of identities. This is the definition that makes lattices algebras in the sense of universal algebra, and it is the one the subject uses.3 min readRead MoreArticle7 Aug 2026The Definition of an Algebra and its TypeThe central definition of the subject: an algebra is a set with a family of finitary operations indexed by a type. Everything that follows is an elaboration of this one idea.3 min readRead MoreArticle7 Aug 2026Subdirect Products and Subdirect EmbeddingsSubalgebras of a direct product that project onto every factor. The construction is weaker than a direct product but available everywhere, and it is the decomposition the subject actually uses.3 min readRead MoreArticle7 Aug 2026Steiner Triple Systems as AlgebrasSteiner triple systems recast as algebras, so that combinatorial questions about them become questions about varieties and congruences.3 min readRead MoreArticle7 Aug 2026Boolean Algebras: Axioms and First ExamplesBoolean algebras as a variety: the axioms, the two-element algebra that generates everything, and the examples that motivate the theory.3 min readRead MoreArticle7 Aug 2026Boolean Powers: Construction and Basic PropertiesThe Boolean power of an algebra by a Boolean algebra: a construction that transfers results about Boolean algebras to arbitrary varieties.3 min readRead MoreArticle7 Aug 2026First-Order Languages and SignaturesThe syntax of first-order logic as universal algebra needs it: languages with function and relation symbols, terms, formulas, and the distinction between free and bound variables.3 min readRead MoreArticle7 Aug 2026The Commutator and the Center: Modern DevelopmentsThe commutator theory for congruence-modular varieties: the generalisation of the group commutator that the source's centre section anticipates.2 min readRead MoreArticle11 Jul 2026Numbers, Fractions, and DecimalsWorking arithmetic for engineering: signed numbers, ratio and proportion, percentage, fractions and reciprocals, the inch–millimetre bridge, continued fractions and change-gear ratios, powers and scientific notation, logarithms, complex numbers, permutations, primes, and the constants worth memorising.11 min readRead MoreGuide6 Aug 2026Multiprecision Integer ArithmeticRepresentation of multiprecision integers, schoolbook and fast multiplication, division, modular reduction strategies, and how to choose the right base ring for a computation.6 min readRead MoreGuide6 Aug 2026The Hermite Normal FormDefinition and uniqueness of the Hermite normal form, algorithms for computing it, entry explosion and its remedies, and its role as the representation of choice for modules and ideals.5 min readRead MoreGuide6 Aug 2026The Subresultant Algorithm, Resultants and DiscriminantsThe subresultant polynomial remainder sequence, the resultant as a determinant and as a product over roots, the discriminant, and their use in elimination and in number field arithmetic.5 min readRead MoreGuide6 Aug 2026Representing Algebraic NumbersThe standard, matrix, conjugate-vector and minimal-polynomial representations of an algebraic number, their relative costs, and how to convert between them.4 min readRead MoreGuide6 Aug 2026Class Numbers of Imaginary Quadratic FieldsComputing class numbers of imaginary quadratic fields by enumeration of reduced forms, by analytic class number formulas, and by modular form methods, with the Gauss class number problem.4 min readRead MoreGuide6 Aug 2026Complex Multiplication and Class FieldsComplex multiplication, isogenies, the relation between CM curves and imaginary quadratic orders, Hilbert and Weber class polynomials, and the CM method for curve construction.5 min readRead MoreGuide6 Aug 2026Compositeness Tests: Fermat and Miller–RabinThe Fermat test and its failure on Carmichael numbers, the strong probable prime test of Miller-Rabin, error bounds, deterministic base sets for bounded ranges, and Baillie-PSW.4 min readRead MoreGuide6 Aug 2026Numerical Tables: KEVOS Sourcing PolicyThe KEVOS two-layer knowledge architecture applied to number theory: durable method in the vault, numeric catalogue data sourced from current authoritative databases such as LMFDB and PARI.5 min readRead MoreGuide6 Aug 2026The Hom Functor and Left ExactnessThe Hom functor in both variables, its covariant and contravariant forms, left exactness, and the precise sense in which it fails to be exact.4 min readRead MoreGuide6 Aug 2026Functors and Their Exactness PropertiesCovariant and contravariant functors, composition, additive functors, faithful and full functors, and the exactness hierarchy that governs derived functors.3 min readRead MoreGuide6 Aug 2026The Ext FunctorExt defined via extensions, via projective resolutions and via injective resolutions, the agreement of the three definitions, and the basic vanishing criteria.4 min readRead MoreGuide6 Aug 2026The Long Exact Homology SequenceThe long exact sequence associated to a short exact sequence of complexes, construction of the connecting homomorphism, naturality, and the standard applications.4 min readRead MoreGuide6 Aug 2026The Künneth FormulaThe Kunneth theorem for complexes over a PID, the short exact sequence with the Tor correction term, splitting, and the hypotheses that are genuinely needed.4 min readRead MoreGuide6 Aug 2026Definition of Group Homology and CohomologyGroup homology and cohomology as derived functors of coinvariants and invariants, equivalently as Tor and Ext over the group ring, with the basic properties and long exact sequences.3 min readRead MoreGuide6 Aug 2026Definition of Lie Algebra CohomologyLie algebra cohomology as Ext over the enveloping algebra, invariants and coinvariants, the explicit description in degrees 0 and 1, and derivations.3 min readRead MoreGuide6 Aug 2026Filtered Differential ObjectsFiltrations of chain complexes, the associated graded object, the spectral sequence of a filtered complex, and the interpretation of the pages.3 min readRead MoreGuide6 Aug 2026Relative Derived FunctorsConstruction of relative derived functors from relative resolutions, their properties, the comparison with absolute derived functors, and worked examples.4 min readRead MoreGuide6 Aug 2026Nilpotent Groups and HomologyThe lower central series, homological characterisation of nilpotent quotients, the Stallings and Stammbach theorems, and applications to group presentations.3 min readRead MoreGuide6 Aug 2026Preliminaries: Sets, Classes, Relations and NotationSets, proper classes, relations, kernels and the notational conventions universal algebra shares with model theory: why the collection of all groups cannot be a set, how relation composition is written, and what the symbol ≈ means as against =.5 min readRead MoreGuide6 Aug 2026Lattice Homomorphisms, Isomorphisms and SublatticesLattice homomorphisms, isomorphisms and sublattices: why every lattice homomorphism is monotone but not conversely, how to recognise a genuine sublattice, and the role of embeddings in the M5/N5 characterisation.5 min readRead MoreGuide6 Aug 2026Subuniverses, Subalgebras and the Generation OperatorSubuniverses, subalgebras, the generation operator Sg(X), and the two equivalent descriptions of generation as intersection from above and as term-closure from below.4 min readRead MoreGuide6 Aug 2026Boolean Rings and the Boolean Algebra-Ring CorrespondenceBoolean rings, the mutual translation with Boolean algebras, term equivalence as the precise relationship, and why the ring picture makes ideals and the prime spectrum available.6 min readRead MoreGuide6 Aug 2026Ultraproducts and Jonsson's LemmaUltraproducts in the algebraic setting, congruence-distributive varieties, Jónsson's lemma and its proof strategy, and the consequences for finitely generated varieties.5 min readRead MoreGuide6 Aug 2026Satisfaction, Truth and Elementary EquivalenceThe satisfaction relation defined by recursion, truth for sentences, theories and models, elementary equivalence, and why elementarily equivalent structures can be non-isomorphic.5 min readRead MoreGuide6 Aug 2026The Seventeen Open Problems: Status Then and NowThe source's seventeen open problems listed in full, grouped by section, with what can be said about their status and an explicit account of where certainty ends.6 min readRead MoreArticle7 Aug 2026Reading Paths: the Short Course and the Research TrackThe source text divides into a short introductory course and a research-oriented remainder. This page sets out both routes explicitly, so that a reader can take the material at the depth they actually need.3 min readRead MoreArticle7 Aug 2026Partial Orders, Posets and BoundsPartial orders, the posets they generate, and the bound notions — upper and lower bounds, suprema and infima — that make the order-theoretic definition of a lattice possible.3 min readRead MoreArticle7 Aug 2026A Catalogue of Algebras: Groups, Rings, LatticesThe standard algebraic structures presented uniformly as algebras, showing how each familiar definition translates into a type plus a set of identities.3 min readRead MoreArticle7 Aug 2026Subdirectly Irreducible AlgebrasThe algebras that admit no non-trivial subdirect decomposition. They are characterised by a single lattice-theoretic condition and serve as the atoms of the structure theory.3 min readRead MoreArticle7 Aug 2026Squags and SloopsThe two varieties of algebras associated with Steiner triple systems — one idempotent, one with a distinguished element — and the relationship between them.3 min readRead MoreArticle7 Aug 2026Boolean Algebra Identities and DualityThe identity calculus of Boolean algebras: De Morgan's laws, involution, absorption, and the duality principle that halves every proof.3 min readRead MoreArticle7 Aug 2026Bounded Boolean Powers and Transfer ResultsThe bounded variant of the Boolean power construction and the transfer theorems it supports.3 min readRead MoreArticle7 Aug 2026First-Order Structures and InterpretationStructures as the semantic counterpart of languages: sets carrying interpretations of every function, constant and relation symbol.2 min readRead MoreArticle7 Aug 2026The Classification of Varieties and Tame Congruence TheoryThe programme of classifying locally finite varieties by the local structure of their finite members, developed after the source text.3 min readRead MoreGuide6 Aug 2026Modular Exponentiation and Powering AlgorithmsLeft-to-right and right-to-left binary powering, sliding window exponentiation, addition chains, and the generic monoid formulation that makes one routine serve many algebraic structures.6 min readRead MoreGuide6 Aug 2026The Smith Normal Form and Its ApplicationsThe Smith normal form, elementary divisors, the structure theorem for finitely generated abelian groups, and the application to class group structure determination.4 min readRead MoreGuide6 Aug 2026Factorisation of Polynomials Modulo a PrimeSquarefree decomposition, distinct-degree factorisation by gcd with x^(q^d) − x, equal-degree splitting by Cantor-Zassenhaus, and Berlekamp's linear-algebra approach.5 min readRead MoreGuide6 Aug 2026Baby-Step Giant-Step and Class Group StructureThe baby-step giant-step method for discrete logarithms and group order, its application to class groups when an approximation to the order is available, and determination of group structure.4 min readRead MoreGuide6 Aug 2026Computing Galois Groups of Number FieldsThe resolvent method for determining Galois groups, the Frobenius cycle-type approach via Dedekind's theorem, transitive group classification by degree, and test polynomials.5 min readRead MoreGuide6 Aug 2026Elliptic Curve L-functions and the Birch–Swinnerton-Dyer ConjectureThe zeta function of a curve, the L-function as an Euler product, modularity and analytic continuation, and the Birch-Swinnerton-Dyer conjecture with its computational uses.4 min readRead MoreGuide6 Aug 2026Classical Primality Proofs: Pocklington and LehmerThe Pocklington-Lehmer N−1 test, partial factorisation requirements, the N+1 test with Lucas sequences, and combined methods.4 min readRead MoreGuide6 Aug 2026Shanks's SQUFOF Factoring MethodThe SQUFOF algorithm: continued fraction expansion of the square root, square forms, the reverse cycle, multipliers, and why it excels in a narrow but important range.4 min readRead MoreGuide6 Aug 2026Direct Sums, Products and Split SequencesDirect sums and direct products of modules, their universal properties, the splitting lemma and equivalent characterisations of split short exact sequences.4 min readRead MoreGuide6 Aug 2026Duality and Opposite CategoriesThe opposite category, the duality principle, dual pairs of notions in homological algebra, and the limits of formal duality.3 min readRead MoreGuide6 Aug 2026Computing Ext GroupsTechniques for computing Ext: choice of variable, use of the long exact sequences, standard computations over the integers, and Ext for cyclic and finitely generated modules.4 min readRead MoreGuide6 Aug 2026Chain Homotopy and the Comparison TheoremChain homotopy between chain maps, homotopy equivalence, the comparison theorem for projective resolutions, and independence of derived functors from the chosen resolution.4 min readRead MoreGuide6 Aug 2026Universal Coefficients and the Dual Künneth FormulaThe universal coefficient theorems for homology and cohomology, the Ext and Tor correction terms, naturality and splitting, and the dual Kunneth formula.3 min readRead MoreGuide6 Aug 2026Low-Dimensional Group CohomologyExplicit descriptions of H^0, H^1 and H_1 for group cohomology, with trivial and non-trivial coefficients, and their interpretations.3 min readRead MoreGuide6 Aug 2026Lie Algebra Extensions and H2Extensions of Lie algebras with abelian kernel, the classification by H^2, central extensions, and examples including the Heisenberg and affine algebras.4 min readRead MoreGuide6 Aug 2026Convergence of Spectral SequencesConvergence conditions for spectral sequences, first-quadrant and bounded cases, conditional convergence, and the failure modes including non-vanishing lim^1.3 min readRead MoreGuide6 Aug 2026SatellitesLeft and right satellites of an additive functor, their construction, the connection with derived functors, and their use when the category lacks enough projectives.4 min readRead MoreGuide6 Aug 2026Finiteness Conditions on GroupsCohomological dimension of groups, the Stallings-Swan theorem, finiteness conditions FP_n and F_n, duality groups, and the role of torsion.4 min readRead MoreGuide6 Aug 2026Distributive and Modular Lattices: the M5 and N5 CriteriaDistributive and modular lattices, the self-duality of both conditions, and the forbidden-sublattice theorems that characterise them by the non-embeddability of M5 and N5.4 min readRead MoreGuide6 Aug 2026The Subalgebra Lattice as an Algebraic LatticeThe subuniverse lattice as an algebraic lattice, the Birkhoff–Frink representation theorem, and why an exact characterisation closes the question for arbitrary algebras while leaving the finite case open.4 min readRead MoreGuide6 Aug 2026Identities and Birkhoff's HSP TheoremIdentities, satisfaction, the operators M and Id as a Galois connection, and Birkhoff's HSP theorem identifying varieties with equationally definable classes, with proof.5 min readRead MoreGuide6 Aug 2026Filters, Ideals and UltrafiltersFilters and ideals on a Boolean algebra, principal and free filters, ultrafilters and their three equivalent characterisations, and the role of ultrafilters throughout the rest of the subject.6 min readRead MoreGuide6 Aug 2026Primal AlgebrasPrimal algebras, Foster's theorem that they generate varieties equivalent to Boolean algebras, the representation of every member as a Boolean power, and the relationship to functional completeness.5 min readRead MoreGuide6 Aug 2026Elementary Substructures and the Lowenheim-Skolem TheoremsElementary substructures and extensions, the Tarski–Vaught test, the downward and upward Löwenheim–Skolem theorems, and the Skolem paradox.6 min readRead MoreGuide6 Aug 2026Tame Congruence TheoryTame congruence theory: minimal sets, the five local types, type sets of varieties, and how the classification refines and largely supersedes the 1981 congruence-condition hierarchy.5 min readRead MoreArticle7 Aug 2026The Prerequisite Dependency GraphThe source carries an explicit diagram of prerequisites showing which sections depend on which. This page renders that dependency structure as navigable text and draws out the consequences for study order.3 min readRead MoreArticle7 Aug 2026Lattices as Posets and the Equivalence TheoremThe order-theoretic definition of a lattice and the theorem establishing that it agrees exactly with the equational definition. Both directions of the construction are given, together with the reason the correspondence is a genuine equivalence.3 min readRead MoreArticle7 Aug 2026Semigroups, Monoids and Quasigroups as AlgebrasThe one-operation structures and the divisibility structures, presented as algebras. Quasigroups in particular require a careful choice of type, and that choice illustrates a general principle.3 min readRead MoreArticle7 Aug 2026Birkhoff's Subdirect Representation TheoremEvery algebra is a subdirect product of subdirectly irreducible algebras. The theorem holds with no hypotheses whatever and is the foundation of the structure theory.3 min readRead MoreArticle7 Aug 2026Quasigroups, Loops and Latin SquaresThe identification of finite quasigroups with Latin squares, and the algebraic vocabulary this supplies for a purely combinatorial object.2 min readRead MoreArticle7 Aug 2026Atoms and Finite Boolean AlgebrasAtoms as the minimal non-zero elements, the classification of finite Boolean algebras as power sets, and the failure of that classification in the infinite case.3 min readRead MoreArticle7 Aug 2026Ultraproducts in Universal AlgebraThe ultraproduct construction from the algebraic side: reduced products modulo an ultrafilter, and the properties they inherit.3 min readRead MoreArticle7 Aug 2026Satisfaction and the Tarski Truth DefinitionThe recursive definition of truth in a structure, and the reason a recursion over formulas requires assignments rather than sentences alone.3 min readRead MoreArticle7 Aug 2026Decidability Questions in Universal AlgebraWhich questions about varieties and algebras admit algorithms, and the dividing lines that later work established.3 min readRead MoreArticle11 Jul 2026Algebra and EquationsRearranging formulas, essential identities, linear and simultaneous equations, quadratics and cubics, Newton–Raphson root finding, arithmetic and geometric progressions, binomial approximations, and the working rules of derivatives and integrals — with machine-shop worked examples.8 min readRead MoreGuide6 Aug 2026The Euclidean Algorithm and GCD ComputationEuclid's algorithm and its worst case, the binary GCD, Lehmer's method for multiprecision operands, and how to choose between them by operand size.6 min readRead MoreGuide6 Aug 2026Lattices and Quadratic FormsLattices as discrete subgroups, bases and unimodular change of basis, the Gram matrix and determinant, successive minima, and the correspondence with positive definite quadratic forms.4 min readRead MoreGuide6 Aug 2026Hensel Lifting and Factorisation over the IntegersHensel's lemma and quadratic lifting, Mignotte's coefficient bounds, the exponential recombination problem, and the LLL-based polynomial-time solution.5 min readRead MoreGuide6 Aug 2026Discriminants and Integral BasesThe field discriminant, integral bases, the index of an equation order, and why computing the maximal order reduces to factoring the polynomial discriminant.5 min readRead MoreGuide6 Aug 2026Real Quadratic Fields and the Infrastructure MethodRegulators and fundamental units of real quadratic fields, the continued fraction algorithm, the exponential cost problem, and Shanks's infrastructure with the distance function.5 min readRead MoreGuide6 Aug 2026Class Group and Unit Computation in General Number FieldsThe general sub-exponential algorithm for class groups and units: factor base selection, ideal reduction, relation collection, the relation matrix and its kernel, and verification against the analytic class number formula.5 min readRead MoreGuide6 Aug 2026Algorithms for Elliptic Curves over ℚComputing minimal Weierstrass models, Tate's algorithm for reduction type and conductor, torsion subgroup determination, heights, and descent for rank computation.5 min readRead MoreGuide6 Aug 2026The Jacobi Sum Primality TestThe Adleman-Pomerance-Rumely test and its Cohen-Lenstra refinement: cyclotomic extensions, characters and Jacobi sums, the basic test, and the final trial division stage.4 min readRead MoreGuide6 Aug 2026Pollard's p−1 Method and Its RelativesThe p−1 method with its two stages, smoothness assumptions, the p+1 method using Lucas sequences, and the implications for choosing cryptographic primes.4 min readRead MoreGuide6 Aug 2026Free and Projective ModulesFree modules and bases, projective modules and the lifting property, the equivalence with direct summands of free modules, and projective resolutions.4 min readRead MoreGuide6 Aug 2026Natural TransformationsNatural transformations and naturality squares, natural isomorphisms, functor categories, and the role of naturality in the long exact sequences.4 min readRead MoreGuide6 Aug 2026The Two Long Exact Sequences of ExtThe long exact sequences of Ext in the first and second variables, their connecting homomorphisms, naturality, and the direction reversal caused by contravariance.3 min readRead MoreGuide6 Aug 2026Projective and Injective ResolutionsConstruction of projective and injective resolutions, the horseshoe lemma, projective and injective dimension, and global dimension of a ring.4 min readRead MoreGuide6 Aug 2026Applications of the Künneth FormulasApplications of the Kunneth and universal coefficient theorems to products of spaces, group cohomology of direct products, and the ring structure on cohomology.4 min readRead MoreGuide6 Aug 2026Derivations and the Semidirect ProductDerivations and principal derivations, the semidirect product, complements and their conjugacy, and the interpretation of H^1 as classifying complements.3 min readRead MoreGuide6 Aug 2026The Chevalley–Eilenberg ResolutionThe Chevalley-Eilenberg resolution of the trivial module, its differential, the resulting cochain complex, and the bound on cohomological dimension.3 min readRead MoreGuide6 Aug 2026The Ladder of an Exact Couple and Rees SystemsThe ladder diagram of an exact couple, Rees systems, and the systematic treatment of filtered chain complexes and their limits.3 min readRead MoreGuide6 Aug 2026Kan ExtensionsLeft and right Kan extensions, their construction by colimits and limits, the adjunction characterisation, and their role in homology of small categories.3 min readRead MoreGuide6 Aug 2026Modular Representation TheoryRepresentation theory in characteristic dividing the group order, failure of semisimplicity, relative projectivity, complexity and support varieties, and the Quillen stratification.4 min readRead MoreGuide6 Aug 2026Complete Lattices and Algebraic LatticesComplete lattices, algebraic lattices, compact elements, and the theorem that a lattice is algebraic exactly when it is the lattice of closed sets of a finitary closure operator.5 min readRead MoreGuide6 Aug 2026The Irredundant Basis TheoremIrredundant bases, the failure of the exchange property outside vector spaces, and the theorem that the set of irredundant basis sizes of a finitely generated algebra has no gaps.4 min readRead MoreGuide6 Aug 2026Equational Logic and the Completeness TheoremThe formal system of equational logic, its five inference rules, soundness, and Birkhoff's completeness theorem identifying derivability with semantic consequence.5 min readRead MoreGuide6 Aug 2026The Boolean Prime Ideal TheoremThe Boolean Prime Ideal Theorem, its equivalent formulations, its strength relative to the axiom of choice, and which results in this collection depend on it.5 min readRead MoreGuide6 Aug 2026Boolean ProductsBoolean products as subdirect products over a Boolean space with a patching condition, the relationship to sheaves, the equaliser condition, and the classes of algebras admitting such representations.5 min readRead MoreGuide6 Aug 2026Reduced Products and Filtered ProductsReduced products over a filter, the congruence they induce, the direct product and ultraproduct as extreme cases, and the Horn sentence preservation theorem.5 min readRead MoreGuide6 Aug 2026The Algebraic Approach to Constraint SatisfactionThe constraint satisfaction problem, polymorphisms as an algebraic invariant, the Feder–Vardi dichotomy conjecture, the algebraic reformulation, and the 2017 resolution.5 min readRead MoreArticle7 Aug 2026Set-Theoretic PreliminariesThe set-theoretic apparatus the subject actually uses: classes as well as sets, indexed families, direct products and powers, and the specific conventions that differ from ordinary practice.3 min readRead MoreArticle7 Aug 2026Lattice Homomorphisms and Order PreservationMaps between lattices that respect the operations, and the sharp distinction between lattice homomorphisms and merely order-preserving maps — a distinction that has no analogue in group or ring theory.3 min readRead MoreArticle7 Aug 2026Modules and R-Modules as AlgebrasModules over a fixed ring as algebras with one unary operation per scalar, and the sense in which modules are the model case for the whole structure theory of congruence-modular varieties.3 min readRead MoreArticle7 Aug 2026Simple Algebras and Congruence SimplicityAlgebras whose only congruences are the two trivial ones. Simplicity is the strongest indecomposability condition and appears throughout the structure theory of varieties.3 min readRead MoreArticle7 Aug 2026Orthogonal Latin Squares and Euler's ConjectureOrthogonality of Latin squares, Euler's 1782 conjecture about which orders admit orthogonal pairs, and the algebraic formulation of the question.3 min readRead MoreArticle7 Aug 2026Boolean Rings and Idempotent RingsRings in which every element is idempotent, their forced properties, and their status as an equationally defined class.2 min readRead MoreArticle7 Aug 2026Jónsson's Lemma for Congruence-Distributive VarietiesThe theorem bounding the subdirectly irreducible algebras of a congruence-distributive variety, and the structural consequences that follow from it.3 min readRead MoreArticle7 Aug 2026Elementary Equivalence and Elementary SubstructuresStructures indistinguishable by first-order sentences, and substructures that agree with the ambient structure on every formula.3 min readRead MoreArticle7 Aug 2026Boolean Constructions: Recent WorkDevelopments in Boolean product representations and discriminator varieties following the period the source describes.3 min readRead MoreGuide6 Aug 2026The Extended Euclidean Algorithm and Modular InversesThe extended Euclidean algorithm, its loop invariants, half-extended variants, modular inversion, and rational reconstruction from a partial remainder sequence.5 min readRead MoreGuide6 Aug 2026Gram–Schmidt OrthogonalisationThe Gram-Schmidt procedure, the mu coefficients and their role in reduction conditions, the numerical instability of the classical algorithm, and exact integral alternatives.4 min readRead MoreGuide6 Aug 2026Root Finding over the Complex NumbersNumerical root finding for polynomials over C: conditioning, the Newton and Aberth methods, splitting-circle approaches, and the precision management needed to support exact number field computations.4 min readRead MoreGuide6 Aug 2026Orders and Ideals in Number FieldsOrders, the maximal order, fractional and integral ideals, unique factorisation of ideals in a Dedekind domain, ideal arithmetic by Hermite normal form, and the two-element representation.4 min readRead MoreGuide6 Aug 2026The Cohen–Lenstra HeuristicsThe Cohen-Lenstra heuristics for class groups of quadratic fields, the weighting principle, predicted divisibility frequencies, and their role in validating computations.4 min readRead MoreGuide6 Aug 2026Elliptic Curve Primality ProvingGoldwasser-Kilian and Atkin-Morain elliptic curve primality proving: the group order downstep, the CM method for avoiding point counting, certificate structure and verification.5 min readRead MoreGuide6 Aug 2026The Continued Fraction Factoring MethodCFRAC: generating small residues from the continued fraction expansion of the square root, smoothness testing, the linear algebra step, and its historical role as precursor to the sieves.4 min readRead MoreGuide6 Aug 2026Projective Modules over a Principal Ideal DomainStructure of projective and finitely generated modules over a PID, the structure theorem, torsion and free parts, and the resulting short projective resolutions.4 min readRead MoreGuide6 Aug 2026Products, Coproducts and Universal ConstructionsUniversal properties, products and coproducts in a general category, initial and terminal objects, and uniqueness up to unique isomorphism.3 min readRead MoreGuide6 Aug 2026The Stein–Serre Theorem for Abelian GroupsThe Stein-Serre theorem characterising countable torsion-free abelian groups that are free, via vanishing of Ext, and related Whitehead-type problems.3 min readRead MoreGuide6 Aug 2026Derived FunctorsLeft and right derived functors, their construction from resolutions, the fundamental properties, and the axiomatic characterisation by universality.4 min readRead MoreGuide6 Aug 2026Cohomology of Finite Cyclic GroupsThe periodic free resolution for a finite cyclic group, the resulting periodic cohomology, norm and difference maps, and Tate cohomology.3 min readRead MoreGuide6 Aug 2026Semisimple Lie Algebras and the Whitehead LemmasThe Killing form and Cartan's criterion, the Casimir element, the two Whitehead lemmas, and their consequences including Weyl's complete reducibility theorem and Levi's theorem.4 min readRead MoreGuide6 Aug 2026Completions of Filtrations and lim1Inverse limits and their failure of exactness, the derived functor lim^1, the Mittag-Leffler condition, the Milnor sequence, and completion of filtered objects.3 min readRead MoreGuide6 Aug 2026Homology of Small CategoriesHomology of a small category with coefficients in a functor, the nerve and its geometric realisation, comparison with group homology, and the associated spectral sequences.3 min readRead MoreGuide6 Aug 2026Derived and Stable CategoriesThe derived category, localisation at quasi-isomorphisms, triangulated structure, derived functors in the modern sense, and the stable module category.4 min readRead MoreGuide6 Aug 2026Equivalence Relations and the Partition LatticeThe lattice of equivalence relations on a set, its identification with the partition lattice, why joins require alternating composites, and permutability as the condition that makes joins easy.5 min readRead MoreGuide6 Aug 2026Congruences and Quotient AlgebrasCongruences as equivalence relations compatible with the operations, the construction of quotient algebras, and why congruences rather than subobjects are the general quotient device.5 min readRead MoreGuide6 Aug 2026Fully Invariant Congruences and Equational TheoriesEquational theories as fully invariant congruences on the term algebra, the dual isomorphism with the lattice of varieties, and equational bases.5 min readRead MoreGuide6 Aug 2026Syntactic Monoids and Kleene's TheoremThe syntactic congruence and syntactic monoid, Kleene's characterisation of recognisable languages as the regular ones, Schützenberger's star-free theorem, and Eilenberg's variety correspondence.6 min readRead MoreGuide6 Aug 2026Stone Duality and Boolean SpacesStone duality: the Stone space of ultrafilters, Boolean spaces as compact Hausdorff totally disconnected spaces, the dual equivalence of categories, and how it seeds the Boolean product machinery.6 min readRead MoreGuide6 Aug 2026Discriminator VarietiesThe ternary discriminator term, discriminator varieties, the Bulman-Fleming–Keimel–Werner representation theorem, and the exceptional package of structural properties that follows.5 min readRead MoreGuide6 Aug 2026The Finite Basis Problem after TarskiThe resolution of Tarski's finite basis problem, the positive finite basis theorems obtained since Baker's, and what the negative answer means for how the question is now approached.5 min readRead MoreArticle7 Aug 2026Relations, Functions and Ordinals: a Working ReferenceA working reference for the relational and functional apparatus the subject assumes: n-ary relations, inverses, relational product, injections and surjections, and the ordinal notation used in transfinite constructions.3 min readRead MoreArticle7 Aug 2026Sublattices and Lattice IsomorphismSubsets of a lattice that are lattices in their own right under the inherited operations, the difference between a sublattice and a sub-poset that happens to be a lattice, and the classification of lattices up to isomorphism.3 min readRead MoreArticle7 Aug 2026Subalgebras and Algebra IsomorphismSubalgebras as subsets closed under the operations, the notion of embedding, and isomorphism as the equivalence under which algebras are classified.3 min readRead MoreArticle7 Aug 2026Class Operators H, S, P and their CompositionThe operators taking homomorphic images, subalgebras and products of a class, the inclusions among their composites, and the identity HSP that computes the generated variety.3 min readRead MoreArticle7 Aug 2026The Refutation of Euler's ConjectureThe 1959–60 disproof of Euler's conjecture by Bose, Shrikhande and Parker, and the algebraic construction that produced the counterexamples.3 min readRead MoreArticle7 Aug 2026The Boolean Algebra / Boolean Ring CorrespondenceThe term-equivalence between Boolean algebras and Boolean rings: each structure's operations are term operations of the other, so the two varieties are the same variety in different notation.3 min readRead MoreArticle7 Aug 2026Primal Algebras and Functional CompletenessFinite algebras whose term operations are all possible operations, and the remarkable rigidity this forces.3 min readRead MoreArticle7 Aug 2026The Tarski–Vaught Test and Löwenheim–SkolemThe practical criterion for recognising elementary substructures and the theorems that build them at prescribed cardinalities.3 min readRead MoreArticle7 Aug 2026Structure Theory and Finite Basis DevelopmentsAdvances in the structure theory of varieties and in the finite basis problem after the source's period.3 min readRead MoreArticle11 Jul 2026GeometryAnalytical and mensuration geometry for engineering: lines and coordinates, polar conversion, the circle and conic sections, areas of plane figures, regular polygon data, circular segments and tank contents, volumes of solids with the prismoidal and Pappus rules, and circles-in-circles roller arrangements.9 min readRead MoreGuide6 Aug 2026The LLL Lattice Reduction AlgorithmThe LLL algorithm: reduction conditions, the swap-and-reduce loop, the potential function that proves polynomial termination, quality guarantees, and the deep-insertion and floating-point variants.5 min readRead MoreGuide6 Aug 2026Decomposition of Prime Numbers in Number FieldsRamification indices and residue degrees, the fundamental identity, Dedekind's theorem relating prime decomposition to polynomial factorisation modulo p, and computing valuations.5 min readRead MoreGuide6 Aug 2026The Elliptic Curve Method (ECM)Lenstra's elliptic curve factoring method: curves modulo a composite, the failed inversion that reveals a factor, stage one and stage two, curve parameterisations, and the role of ECM in practice.5 min readRead MoreGuide6 Aug 2026Injective Modules and DualizationInjective modules, the extension property, Baer's criterion, divisible groups, and the existence of enough injectives.4 min readRead MoreGuide6 Aug 2026Pullbacks, Pushouts and General LimitsPullbacks and pushouts, their construction in module categories, limits and colimits over a diagram, and exactness of filtered colimits.4 min readRead MoreGuide6 Aug 2026The Tensor Product of ModulesThe tensor product by universal property and construction, right exactness, flat modules, and the tensor-hom adjunction.4 min readRead MoreGuide6 Aug 2026The Long Exact Sequences of Derived FunctorsDerivation of the long exact sequences of derived functors via the horseshoe lemma, their naturality, dimension shifting and the standard computational patterns.3 min readRead MoreGuide6 Aug 2026Resolutions for Group CohomologyThe bar resolution and its normalised version, cocycles and coboundaries in explicit terms, resolutions from presentations, and the trade-off between explicitness and size.3 min readRead MoreGuide6 Aug 2026Hilbert's Syzygy TheoremHilbert's chain-of-syzygies theorem, the Koszul resolution, global dimension of polynomial rings, and the historical and modern significance.3 min readRead MoreGuide6 Aug 2026The Grothendieck Spectral SequenceThe Grothendieck spectral sequence for a composite of functors, the acyclicity hypothesis, the five-term sequence and edge maps, and the standard specialisations.3 min readRead MoreGuide6 Aug 2026Closure Operators and Galois ConnectionsClosure operators and closure systems, the correspondence with complete lattices, the finitary case and algebraic lattices, and Galois connections as the standard source of closure operators in algebra and logic.5 min readRead MoreGuide6 Aug 2026The Congruence Lattice Con AThe congruence lattice: completeness, algebraicity, its position inside Eq(A), simple and subdirectly irreducible algebras read off from it, and the classification of varieties by congruence conditions.4 min readRead MoreGuide6 Aug 2026Mal'cev Conditions I: Congruence PermutabilityMal'cev conditions in general, the ternary Mal'cev term characterising congruence permutability, and why term conditions are the right way to classify varieties.5 min readRead MoreGuide6 Aug 2026The Compactness Theorem and its ConsequencesThe compactness theorem, its proof by ultraproducts, the standard consequences including non-standard models, and the systematic list of properties first-order logic cannot express.5 min readRead MoreGuide6 Aug 2026Universal Algebra: Computation and SourcesThe sourcing policy for this collection: what is written here, what is deliberately not transcribed, the reasoning behind the split, and where to obtain current computational results and literature.6 min readRead MoreArticle7 Aug 2026Notation and ConventionsA consolidated reference for the notation used throughout the subject, drawn from the source's own special-notation tables and organised by what the symbol is for rather than by where it first appears.4 min readRead MoreArticle7 Aug 2026Distributive Lattices and their CharacterisationThe distributive law for lattices, its self-dual character, and the concrete examples that make distributivity the most important special condition in lattice theory.3 min readRead MoreArticle7 Aug 2026Subuniverses and the Generation Operator SgThe operator Sg that produces the smallest subuniverse containing a given set, its two equivalent descriptions, and the finitary character that makes it an algebraic closure operator.3 min readRead MoreArticle7 Aug 2026Varieties and the Variety Generated by a ClassVarieties as classes closed under H, S and P, the variety generated by a class, and the lattice of subvarieties.3 min readRead MoreArticle7 Aug 2026Finite State Acceptors and Recognisable LanguagesFinite automata presented as algebras with unary operations, and the languages they recognise, giving an algebraic route into the theory of regular languages.3 min readRead MoreArticle7 Aug 2026Filters and Ideals in Boolean AlgebrasFilters as upward-closed meet-closed subsets, ideals as their duals, and their correspondence with congruences.3 min readRead MoreArticle7 Aug 2026The Primal Algebra Characterisation TheoremThe theorem identifying primal algebras by intrinsic conditions, and the representation of the generated variety by Boolean powers.3 min readRead MoreArticle7 Aug 2026Theories, Models and AxiomatisabilityTheories as sets of sentences, model classes, and the question of which classes of algebras are first-order axiomatisable.3 min readRead MoreArticle7 Aug 2026Applications to Computer Science and Model TheoryThe two application areas the source identifies, and what became of them.3 min readRead MoreGuide6 Aug 2026Continued Fraction ExpansionsSimple continued fractions, convergent recurrences and best-approximation properties, Lagrange's periodicity theorem, and the expansion of a square root used for Pell's equation and factoring.4 min readRead MoreGuide6 Aug 2026Applications of the LLL AlgorithmPractical applications of lattice reduction: integer kernel and image computation, integer relation detection, recovering minimal polynomials from numerical approximations, and simultaneous Diophantine approximation.5 min readRead MoreGuide6 Aug 2026Class Groups, Units and the RegulatorThe ideal class group, Dirichlet's unit theorem, the regulator, Minkowski's bound, and the analytic class number formula used to verify computed values.4 min readRead MoreGuide6 Aug 2026The Quadratic Sieve and MPQSThe quadratic sieve, sieving by roots of a polynomial, the multiple polynomial variation, large prime variations, and the linear algebra stage.4 min readRead MoreGuide6 Aug 2026Injective Modules over a Principal Ideal DomainInjective modules over a PID characterised by divisibility, the classification of injective abelian groups, and the resulting short injective resolutions.3 min readRead MoreGuide6 Aug 2026Adjoint FunctorsAdjoint pairs, unit and counit, the tensor-hom adjunction, preservation of limits and colimits, and the exactness consequences that make adjointness central to homological algebra.4 min readRead MoreGuide6 Aug 2026The Tor FunctorTor defined by projective resolutions, symmetry in its two variables, vanishing criteria, and the standard computations over the integers.4 min readRead MoreGuide6 Aug 2026Ext via Projective and Injective ResolutionsComputing Ext by resolving either variable, the double complex proof that the two agree, and the practical consequences for choosing a computation.4 min readRead MoreGuide6 Aug 2026Group Extensions and H2Group extensions with abelian kernel, factor sets and their equivalence, the classification by H^2, central extensions, and the obstruction interpretation.4 min readRead MoreGuide6 Aug 2026The Lyndon–Hochschild–Serre Spectral SequenceThe LHS spectral sequence for a normal subgroup, its E2 page, the action of the quotient, the five-term sequence, and worked applications.3 min readRead MoreGuide6 Aug 2026Homomorphisms and the Isomorphism TheoremsHomomorphisms, kernels, the first, second and third isomorphism theorems in arbitrary type, the correspondence theorem, and the congruence extension property.4 min readRead MoreGuide6 Aug 2026Mal'cev Conditions II: Congruence Distributivity and Jonsson TermsCongruence-distributive varieties, the Jónsson term characterisation, Jónsson's lemma, and congruence modularity with Day terms.5 min readRead MoreGuide6 Aug 2026Preservation Theorems: Horn, Universal and Positive SentencesThe preservation theorems: Łoś–Tarski for substructures, Lyndon for homomorphisms, Keisler–Galvin for reduced products, Chang–Łoś–Suszko for unions of chains, and their relationship to Birkhoff's theorem.5 min readRead MoreArticle7 Aug 2026Modular Lattices and the Modular LawThe modular law as a conditional weakening of distributivity, Dedekind's theorem that subgroup lattices of abelian groups are modular, and the reason modularity is the right hypothesis for a large part of algebra.3 min readRead MoreArticle7 Aug 2026The Subalgebra Lattice Sub(A) is AlgebraicThe theorem that the subuniverses of any algebra form an algebraic lattice, and the converse showing that every algebraic lattice arises this way.3 min readRead MoreArticle7 Aug 2026Terms and the Term Algebra T(X)Terms as formal expressions built from variables and operation symbols, the term algebra they form, and its absolute freeness.3 min readRead MoreArticle7 Aug 2026The Syntactic Monoid and Kleene's TheoremThe monoid canonically associated with a language, Kleene's characterisation of the recognisable languages as the regular ones, and the algebraic classification programme this opens.3 min readRead MoreArticle7 Aug 2026Ultrafilters and the Boolean Prime Ideal TheoremMaximal proper filters, their characterisation by the decision property, and the existence theorem that underwrites Stone duality and the ultraproduct construction.3 min readRead MoreArticle7 Aug 2026Boolean Products: Definition and MotivationA representation of an algebra as continuous sections over a Boolean space, replacing sheaf theory with a simpler and equally powerful formulation.3 min readRead MoreArticle7 Aug 2026Filters, Reduced Products and the ConstructionThe reduced product construction in detail: how a filter on the index set determines which coordinates matter.3 min readRead MoreArticle7 Aug 2026The Seventeen Open Problems: a Status RegisterA status register for the open problems stated in the source's closing chapter, reporting what is known rather than asserting resolutions.3 min readRead MoreGuide6 Aug 2026Legendre, Jacobi and Kronecker SymbolsQuadratic residues, Euler's criterion, the Legendre symbol, its Jacobi and Kronecker extensions, and the reciprocity-based algorithm that evaluates them in logarithmic time.4 min readRead MoreGuide6 Aug 2026The Number Field SieveThe number field sieve: polynomial selection, sieving over two sides, the algebraic factor base and character columns, the square root step in a number field, and the special number field sieve.5 min readRead MoreGuide6 Aug 2026Cofree Modules and Essential ExtensionsCofree modules via Hom from the ring, the adjunction producing enough injectives, essential extensions, and the existence and uniqueness of injective hulls.4 min readRead MoreGuide6 Aug 2026Abelian CategoriesAdditive and abelian categories, the axioms, exactness in a general abelian category, the Freyd-Mitchell embedding theorem, and the standard examples.4 min readRead MoreGuide6 Aug 2026Yoneda Ext and n-Fold ExtensionsThe Yoneda description of Ext^n by n-fold extensions, the equivalence relation, splicing as composition, and the Yoneda product.4 min readRead MoreGuide6 Aug 2026H2, Hopf's Formula and the Schur MultiplierHopf's formula for H_2 in terms of a free presentation, the Schur multiplier, universal central extensions, and the connection to the lower central series.3 min readRead MoreGuide6 Aug 2026Direct Products, Factor Congruences and Direct IndecomposabilityDirect products, projection homomorphisms, factor congruences as the internal signature of a decomposition, directly indecomposable algebras and the limits of unique factorisation.4 min readRead MoreGuide6 Aug 2026The Center of an Algebra and Affine RepresentationThe centre of an arbitrary algebra, its first-order definition, the characterisation of algebras with Z(A) = ∇ as polynomially equivalent to modules, and the commutator programme it opened.5 min readRead MoreGuide6 Aug 2026Skew-free AlgebrasSkew congruences in direct products, skew-free algebras and varieties, the relationship to congruence permutability and distributivity, and the role in Boolean representations.5 min readRead MoreArticle7 Aug 2026The M5 and N5 Forbidden-Sublattice TheoremsTwo five-element lattices decide modularity and distributivity by exclusion. These theorems convert conditions stated as identities into a finite, checkable structural test.3 min readRead MoreArticle7 Aug 2026Irredundant Bases and the Irredundant Basis TheoremMinimal generating sets that admit no proper generating subset, and the theorem constraining the possible sizes of such bases in an algebra with an algebraic closure operator.3 min readRead MoreArticle7 Aug 2026Term Operations and Polynomial OperationsThe functions terms induce on an algebra, the polynomials obtained by allowing parameters, and the clones these families form.3 min readRead MoreArticle7 Aug 2026Applied Universal Algebra: a SynthesisWhat the two showcase applications have in common, the general recipe they illustrate, and an assessment of the source's prediction about the field's direction.3 min readRead MoreArticle7 Aug 2026Maximal Filters and Boolean CongruencesThe lattice isomorphism between filters and congruences, what maximal filters say about simple quotients, and the resulting proof that 2 is the only subdirectly irreducible Boolean algebra.3 min readRead MoreArticle7 Aug 2026Weak Boolean Products and Patchwork PropertiesThe relaxation of the Boolean product conditions that makes representations available more widely, and what is lost by the relaxation.3 min readRead MoreArticle7 Aug 2026Ultraproducts and Łoś's TheoremThe theorem that an ultraproduct satisfies a sentence exactly when a large set of factors does — the single most useful result in model-theoretic algebra.3 min readRead MoreArticle7 Aug 2026Bibliography and Further Reading GuideA guide to the source's bibliography and to the standard references for the subject, organised by what each is for.3 min readRead MoreGuide6 Aug 2026Square Roots Modulo a PrimeComputing modular square roots: the p ≡ 3 (mod 4) shortcut, the Tonelli–Shanks algorithm and its 2-adic structure, and Cornacchia's algorithm for x² + dy² = p.4 min readRead MoreGuide6 Aug 2026Change of RingsRestriction and extension of scalars, induced and coinduced modules, the change-of-rings theorems, and the associated spectral sequences.3 min readRead MoreGuide6 Aug 2026The Five-Term Exact SequenceThe five-term exact sequence for a normal subgroup, inflation and restriction maps, transgression, and its derivation from the Lyndon-Hochschild-Serre spectral sequence.3 min readRead MoreGuide6 Aug 2026Subdirect Products and Birkhoff's Subdirect Representation TheoremSubdirect products, subdirect irreducibility, the monolith, and Birkhoff's subdirect representation theorem with its Zorn's lemma proof.4 min readRead MoreGuide6 Aug 2026Semisimple VarietiesSemisimple varieties, the relationship to discriminator varieties, residual smallness, and the structure available when subdirect decomposition terminates in simple algebras.4 min readRead MoreGuide6 Aug 2026Three Finite Basis TheoremsThe finite basis problem, Baker's theorem for congruence-distributive varieties, and the further finite basis results in the source, with the proof strategy via bounded subdirectly irreducibles.5 min readRead MoreArticle7 Aug 2026Complete Lattices and Completeness CriteriaLattices in which every subset — not merely every pair — has a supremum and an infimum, and the surprisingly economical criterion that establishes completeness from one half of the condition alone.3 min readRead MoreArticle7 Aug 2026Congruences and the Substitution PropertyCongruences: the equivalence relations compatible with the operations. The substitution property, why it is the right condition, and the failure modes when it is absent.3 min readRead MoreArticle7 Aug 2026Free Algebras and the Universal Mapping PropertyFree algebras in a class, their construction as quotients of the term algebra, and the universal mapping property that characterises them.3 min readRead MoreArticle7 Aug 2026Boolean Spaces and Stone SpacesThe topological spaces that arise as duals of Boolean algebras: compact, Hausdorff, totally disconnected, with a basis of clopen sets.3 min readRead MoreArticle7 Aug 2026The Spectrum of an AlgebraThe spectrum as the Boolean space indexing an algebra's canonical decomposition, and the spectrum of a variety.3 min readRead MoreArticle7 Aug 2026The Compactness Theorem via UltraproductsThe theorem that a finitely satisfiable theory has a model, proved algebraically by an ultraproduct construction, and its consequences.2 min readRead MoreArticle11 Jul 2026Solution of TrianglesEngineering trigonometry reference: functions of angles, degree–radian conversion, right-angled and oblique triangle solutions, identities, computed function tables, interpolation, versed and involute functions, compound angles, and spherical trigonometry, with worked metric examples.24 min readRead MoreGuide6 Aug 2026Solving Polynomial Equations Modulo pRoot-finding modulo a prime: the gcd with x^p − x, equal-degree splitting by random shifts, and the special low-degree cases that admit closed forms.4 min readRead MoreGuide6 Aug 2026Subgroups: Restriction, Corestriction and TransferRestriction and corestriction (transfer) maps, the index formula, consequences for torsion, and detection on Sylow subgroups.3 min readRead MoreGuide6 Aug 2026Class Operators H, S, P and the Definition of a VarietyThe class operators I, S, H, P and P_S, the inclusions and idempotency relations among their composites, and the identification of HSP as the variety-generating closure operator.5 min readRead MoreGuide6 Aug 2026Directly Representable VarietiesDirectly representable varieties, McKenzie's theorem that they are congruence-permutable, the classification of their directly indecomposable members, and the connection to the (discriminator) ⊗ (modular Abelian) decomposition.5 min readRead MoreArticle7 Aug 2026Equivalence Relations and the Partition Lattice Eq(A)Equivalence relations on a set, their equivalent description as partitions, and the complete lattice Eq(A) they form — the ambient lattice inside which every congruence lattice sits.3 min readRead MoreArticle7 Aug 2026Quotient Algebras and the Natural MapThe construction of the quotient algebra modulo a congruence, the natural surjection onto it, and the universal property that makes quotients the right notion.3 min readRead MoreArticle7 Aug 2026Identities, Satisfaction and Equational ClassesIdentities as pairs of terms, what it means for an algebra to satisfy one, and the Galois connection between classes of algebras and sets of identities.3 min readRead MoreArticle7 Aug 2026The Stone Representation TheoremEvery Boolean algebra is isomorphic to a field of sets. The theorem, its proof from the prime ideal theorem, and what it does and does not deliver.3 min readRead MoreArticle7 Aug 2026The Ternary Discriminator FunctionThe operation that tests equality and branches, and the reason it is the single most consequential term operation in the subject.3 min readRead MoreArticle7 Aug 2026Preservation Theorems for Universal SentencesThe theorems matching syntactic form to closure under algebraic constructions, with universal sentences and substructures as the model case.3 min readRead MoreGuide6 Aug 2026Integer Square Roots and Perfect Power DetectionComputing exact integer square roots by Newton's method, fast rejection of non-squares by modular filters, and detection of perfect powers and prime powers.5 min readRead MoreGuide6 Aug 2026Cohomology of Products and Coproducts of GroupsCohomology of a direct product via Kunneth, cohomology of a free product as a direct sum, Mayer-Vietoris for amalgamated products, and the contrast between the two constructions.3 min readRead MoreArticle7 Aug 2026Algebraic Lattices and Compact ElementsCompact elements, algebraic lattices, and the theorem that the subuniverse and congruence lattices of any algebra are algebraic — the structural fact that finitary arity buys.3 min readRead MoreArticle7 Aug 2026The Congruence Lattice Con(A) and its AlgebraicityCon(A) as a complete algebraic lattice, its relationship to the ambient lattice of equivalence relations, and the sense in which it is the fundamental invariant of an algebra.3 min readRead MoreArticle7 Aug 2026Birkhoff's HSP TheoremThe central theorem of universal algebra: a class of algebras is definable by identities exactly when it is closed under homomorphic images, subalgebras and direct products.3 min readRead MoreArticle7 Aug 2026Stone Duality for Boolean AlgebrasThe full categorical duality between Boolean algebras and Boolean spaces: objects correspond, morphisms correspond with reversed direction, and every construction on one side has a counterpart on the other.2 min readRead MoreArticle7 Aug 2026Discriminator Varieties and their StructureVarieties generated by classes of algebras sharing a discriminator term, and the representation theory that makes them, in the source's words, remarkably well-behaved.3 min readRead MoreArticle7 Aug 2026Horn Sentences and Reduced-Product PreservationThe syntactic class matching closure under reduced products, and why direct products behave so well for the classes algebra cares about.3 min readRead MoreArticle11 Jul 2026MatricesMatrix operations for engineering: addition and scalar multiples, matrix multiplication, transpose, determinants, minors, cofactors and the adjoint, rank and singularity, the inverse, solving simultaneous equations by Cramer's rule and by inversion, and rotation matrices for coordinate work.6 min readRead MoreArticle7 Aug 2026Closure Operators and Algebraic ClosureClosure operators, the complete lattices of closed sets they generate, and the correspondence that makes them the organising device behind subuniverses, congruences and generated substructures alike.3 min readRead MoreArticle7 Aug 2026Principal and Generated CongruencesThe congruence generated by a set of pairs, the principal congruences generated by a single pair, and the reason principal congruences are the compact building blocks of Con(A).3 min readRead MoreArticle7 Aug 2026Mal'cev Conditions and Congruence PermutabilityConditions on a variety expressed by the existence of terms satisfying prescribed identities, and Mal'cev's theorem characterising congruence permutability by a single ternary term.3 min readRead MoreArticle7 Aug 2026Clopen Sets and the Duality DictionaryWorking with the duality in practice: how to translate a specific problem into its dual form, and the properties that correspond on each side.3 min readRead MoreArticle7 Aug 2026Quasiprimal Algebras and Pixley's TheoremFinite algebras whose term operations are exactly those preserving the internal isomorphisms, and Pixley's characterisation of them.3 min readRead MoreArticle7 Aug 2026Principal Congruence FormulasFirst-order formulas defining membership in principal congruences uniformly across a variety, and the finiteness conditions that make them available.3 min readRead MoreArticle7 Aug 2026The Congruence Extension PropertyThe condition that congruences on a subalgebra extend to the whole algebra, the varieties that satisfy it, and its role in transferring structural results.3 min readRead MoreArticle7 Aug 2026Congruence-Distributive and Congruence-Modular VarietiesVarieties in which every congruence lattice is distributive or modular, the term conditions characterising them, and the structure theory each unlocks.3 min readRead MoreArticle7 Aug 2026Arithmetical Varieties and Pixley TermsVarieties that are both congruence-permutable and congruence-distributive, and the single term that characterises the combination.3 min readRead MoreArticle7 Aug 2026Sizes of Subdirectly Irreducible AlgebrasBounding how large the subdirectly irreducible members of a variety can be, and the compactness arguments that produce the bounds.3 min readRead MoreArticle11 Jul 2026Manufacturing Data AnalysisStatistics for the shop floor: samples and populations, mean and standard deviation, distribution curves, the normal curve and z-values with a computed table, estimating the fraction outside limits, minimum number of tests, comparing two averages, and the machinability–hardness relationship.6 min readRead MoreArticle7 Aug 2026Homomorphisms, Kernels and the First Isomorphism TheoremHomomorphisms as structure-preserving maps, kernels as congruences, and the first isomorphism theorem in the generality where it belongs.3 min readRead MoreArticle7 Aug 2026The Center of an AlgebraThe centre of a general algebra, defined by a term condition generalising the group centre, and its use in characterising modules up to polynomial equivalence.3 min readRead MoreArticle7 Aug 2026Functionally Complete AlgebrasAlgebras whose polynomial operations exhaust all operations, and the difference parameters make.3 min readRead MoreArticle7 Aug 2026The First Two Finite Basis TheoremsTwo results giving conditions under which a variety has a finite equational basis, and the general shape of finite basis arguments.3 min readRead MoreArticle7 Aug 2026The Second and Third Isomorphism TheoremsThe remaining isomorphism theorems in their general algebraic form, and the hypotheses each requires — including the one that fails without congruence permutability.2 min readRead MoreArticle7 Aug 2026Equational Logic and the Rules of DeductionA formal proof system for identities, its five rules, and the completeness theorem matching syntactic derivability with semantic consequence.3 min readRead MoreArticle7 Aug 2026Skew-Free Algebras and IndependenceAlgebras whose congruences on a product decompose into products of congruences on the factors, and the independence conditions that force this.3 min readRead MoreArticle7 Aug 2026Baker's Finite Basis TheoremThe theorem that every finitely generated congruence-distributive variety of finite type has a finite equational basis — the deepest result in the source's final chapter.3 min readRead MoreArticle11 Jul 2026Engineering EconomicsMoney over time for engineering decisions: simple and compound interest, nominal and effective rates, cash-flow diagrams, equivalence factors with a computed 8% table, net present value, capitalised cost and EUAC, rate of return and payback, benefit–cost ratio, depreciation schedules, break-even analysis, and machine-hour overhead rates.8 min readRead MoreArticle7 Aug 2026The Correspondence Theorem for AlgebrasThe bijection between congruences above a fixed congruence and congruences on the quotient — a lattice isomorphism that makes Con of a quotient an interval in Con of the original.3 min readRead MoreArticle7 Aug 2026Fully Invariant Congruences and CompletenessFully invariant congruences on the term algebra, their correspondence with equational theories, and the lattice anti-isomorphism between theories and varieties.3 min readRead MoreArticle7 Aug 2026Semisimple and Directly Representable VarietiesVarieties whose subdirectly irreducible members are all simple, and those whose finite members are all direct products from a fixed finite list.3 min readRead MoreArticle7 Aug 2026Semantic Embeddings and UndecidabilityInterpreting one theory inside another to transfer undecidability, and the undecidability results this yields for algebraic theories.3 min readRead MoreArticle7 Aug 2026Direct Products and Factor CongruencesThe direct product construction, the projection homomorphisms, and factor congruences — the congruence-lattice signature that detects when an algebra decomposes as a product.3 min readRead MoreArticle7 Aug 2026Directly Indecomposable AlgebrasAlgebras that admit no non-trivial direct decomposition, the congruence-lattice condition characterising them, and the limits of unique factorisation.3 min readRead MoreArticle6 Aug 2026Divisibility, Ideals and Unique FactorizationHow divisibility, division with remainder, ideals, greatest common divisors and the fundamental theorem of arithmetic fit together — the structural foundation for every modular and cryptographic algorithm.10 min readRead MoreArticle6 Aug 2026Euler's Phi Function and Fermat's Little TheoremEuler's totient function, its multiplicativity, Fermat's little theorem and Euler's theorem, the structure of the unit group ℤ*n, Carmichael's lambda and the primitive root theorem.8 min readRead MoreArticle6 Aug 2026The Distribution of PrimesChebyshev bounds, Bertrand's postulate, Mertens' theorems, the prime number theorem and primes in arithmetic progressions — the density results that make random prime generation practical.9 min readRead MoreArticle6 Aug 2026Abelian Groups and Cyclic StructureAbelian groups, subgroups, cosets and quotient groups, Lagrange's theorem, homomorphisms and isomorphism theorems, cyclic groups and the structure theorem for finite abelian groups.9 min readRead MoreArticle6 Aug 2026Rings, Ideals and Quotient RingsCommutative rings with unity, units and zero divisors, integral domains and fields, ideals and quotient rings, ring homomorphisms and the isomorphism theorems, with the CRT as a ring decomposition.7 min readRead MoreArticle6 Aug 2026Polynomial Rings and Unique FactorizationThe ring F[X] as a Euclidean domain, division with remainder, roots and degree bounds, irreducibility, unique factorization, Gauss's lemma for ℤ[X], and polynomial congruences and quotient algebras.7 min readRead MoreArticle6 Aug 2026Finite Fields: Existence, Uniqueness and StructureWhy finite fields have prime-power order, how they are constructed and why they are unique, the subfield lattice, conjugates, norms and traces, and the Frobenius automorphism.7 min readRead MoreArticle6 Aug 2026Quadratic Residues and Quadratic ReciprocityQuadratic residues modulo a prime and a composite, Euler's criterion, the Legendre and Jacobi symbols, the law of quadratic reciprocity and its supplements, and the quadratic residuosity assumption.7 min readRead MoreArticle6 Aug 2026Modules, Vector Spaces and MatricesModules over a ring, vector spaces and dimension, linear independence and bases, matrices as linear maps, rank and kernel, and Gaussian elimination over a field with its algorithmic applications.8 min readRead MoreArticle6 Aug 2026Discrete Probability for Algorithm AnalysisFinite probability distributions, conditional probability and independence, random variables, expectation and variance, Chebyshev and Chernoff bounds, the birthday paradox, hash functions and statistical distance.8 min readRead MoreArticle6 Aug 2026Linearly Generated Sequences and Sparse Linear SystemsLinearly generated sequences and their minimal polynomials, the Berlekamp–Massey and extended-Euclidean methods of computing them, and Wiedemann's algorithm for solving huge sparse linear systems over a finite field.3 min readRead MoreArticle6 Aug 2026Asymptotic Notation and Machine ModelsBig-O, Omega and Theta notation, the RAM model, bit complexity versus operation counts, input size measured in bits, polynomial versus subexponential versus exponential time, and how to state a cost model correctly.8 min readRead MoreArticle6 Aug 2026Euclid's Algorithm and Modular ComputationThe Euclidean and extended Euclidean algorithms with complexity analysis, computing modular inverses, Chinese remaindering in practice, multi-modular computation, and rational reconstruction with its applications.9 min readRead MoreArticle6 Aug 2026Probabilistic AlgorithmsModels of randomised computation, one-sided and two-sided error, error reduction by repetition, generating random numbers in a range without bias, rejection sampling, and generating random primes and factored numbers.7 min readRead MoreArticle6 Aug 2026Probabilistic Primality TestingTrial division, the Fermat test and its failure on Carmichael numbers, the Miller–Rabin test with its 1/4 error bound, practical prime generation, perfect power testing and the equivalence of factoring with computing Euler's phi.8 min readRead MoreArticle6 Aug 2026Deterministic Primality Testing: the AKS AlgorithmThe AKS primality test — the polynomial identity it is based on, the algorithm and its correctness argument, the role of the auxiliary parameter r, complexity, later improvements and why it is not used in practice.7 min readRead MoreArticle6 Aug 2026Generators and Discrete Logarithms in ℤ*pFinding a generator of ℤ*p, generic discrete logarithm algorithms including baby-step giant-step and Pollard rho, the Pohlig–Hellman reduction, and the Diffie–Hellman key establishment protocol.8 min readRead MoreArticle6 Aug 2026Computing Modular Square RootsAlgorithms for the Jacobi symbol, testing quadratic residuosity, extracting square roots modulo a prime with Tonelli–Shanks and Cipolla, lifting to prime powers by Hensel's lemma, and roots modulo a composite.7 min readRead MoreArticle6 Aug 2026Subexponential Factoring and Index CalculusSmooth number densities, the index calculus method for discrete logarithms, the quadratic sieve and number field sieve for factoring, practical improvements, and what record computations imply for key sizes.9 min readRead MoreArticle6 Aug 2026Polynomial Arithmetic and ApplicationsBasic and fast polynomial arithmetic, the Euclidean algorithm over F[X], modular inverses and Chinese remaindering for polynomials, evaluation and interpolation, rational function reconstruction and Reed–Solomon decoding.7 min readRead MoreArticle6 Aug 2026Factoring Polynomials over Finite FieldsSquarefree decomposition, distinct-degree and equal-degree factorization, the Cantor–Zassenhaus algorithm, Berlekamp's linear-algebra method, irreducibility testing and constructing irreducible polynomials.8 min readRead MoreArticle6 Aug 2026The RSA CryptosystemRSA key generation, encryption and signing, the correctness proof from Euler's theorem, why textbook RSA is insecure, padding schemes, known attacks including small exponent and fault attacks, and implementation requirements.7 min readRead MoreArticle7 Aug 2026Computational Number Theory and Algebra: Field OverviewHow number theory, abstract algebra and algorithm analysis combine into a single computational discipline, and how the KEVOS Mathematics library is organised.5 min readRead MoreArticle7 Aug 2026Mathematical Notation and Standing ConventionsThe notational conventions, symbol set and standing assumptions used throughout the KEVOS computational number theory collection.3 min readRead MoreArticle7 Aug 2026Useful Facts and Standard EstimatesThe analytic inequalities, series estimates and elementary bounds relied on repeatedly in the analysis of number-theoretic algorithms.3 min readRead MoreArticle7 Aug 2026Learning Pathways in Computational Number TheorySuggested routes through the collection for cryptography, computer algebra, coding theory and pure mathematics readers.3 min readRead MoreArticle7 Aug 2026Divisibility and PrimalityDivisibility of integers, the definition of primes and composites, and the basic properties that support every later result in elementary number theory.4 min readRead MoreArticle7 Aug 2026Division with Remainder for IntegersThe division algorithm for integers, the uniqueness of quotient and remainder, and the role of well-ordering in establishing it.3 min readRead MoreArticle7 Aug 2026Ideals and Greatest Common Divisors of IntegersGreatest common divisors defined through ideals, Bezout's identity, and why the ideal-theoretic view is the one that generalises.4 min readRead MoreArticle7 Aug 2026Unique Factorization of the IntegersThe fundamental theorem of arithmetic: existence and uniqueness of prime factorisation, and why the uniqueness half is the difficult one.3 min readRead MoreArticle7 Aug 2026Consequences of Unique FactorizationWhat follows from the fundamental theorem: gcd and lcm via exponents, irrationality proofs, divisor counting and multiplicative structure.3 min readRead MoreArticle7 Aug 2026Congruences and Modular ArithmeticCongruence as an equivalence relation compatible with arithmetic, and the basic manipulation rules including where cancellation fails.3 min readRead MoreArticle7 Aug 2026Solving Linear CongruencesSolving ax = b (mod n): the solvability criterion, the exact number of solutions, and the algorithm via extended Euclid.3 min readRead MoreArticle7 Aug 2026Residue Classes and the Ring of Integers Modulo nThe ring Z_n of residue classes, its units and zero divisors, and the condition under which it is a field.4 min readRead MoreArticle7 Aug 2026The Chinese Remainder TheoremThe Chinese remainder theorem as a ring isomorphism, its constructive proof, and its role in decomposing modular computation.3 min readRead MoreArticle7 Aug 2026Euler's Phi FunctionEuler's totient function: its definition, multiplicativity, closed form from the prime factorisation, and computational status.3 min readRead MoreArticle7 Aug 2026Fermat's Little Theorem and Euler's TheoremFermat's little theorem, Euler's generalisation, and their role as the foundation of primality testing and public-key cryptography.3 min readRead MoreArticle7 Aug 2026Arithmetic Functions and Mobius InversionMultiplicative arithmetic functions, Dirichlet convolution, the Mobius function and the inversion formula.3 min readRead MoreArticle7 Aug 2026Asymptotic Notation for Algorithm AnalysisBig-O, Omega, Theta and little-o notation, the conventions that make them precise, and the pitfalls of using them carelessly.4 min readRead MoreArticle7 Aug 2026Machine Models and Complexity TheoryThe random access machine and Turing machine models, what counts as a primitive operation, and how the choice of model affects stated complexities.3 min readRead MoreArticle7 Aug 2026Representing Large IntegersPositional representation of multiprecision integers, base selection, sign handling and normalisation invariants.3 min readRead MoreArticle7 Aug 2026Integer Addition and SubtractionMultiprecision addition and subtraction: carry and borrow propagation, sign handling, and why both are linear.3 min readRead MoreArticle7 Aug 2026Integer MultiplicationSchoolbook multiplication of multiprecision integers, the accumulator requirement, and where the quadratic cost comes from.3 min readRead MoreArticle7 Aug 2026Integer Division with RemainderMultiprecision division: the normalisation step, digit estimation, correction, and why division is harder to implement than multiplication.3 min readRead MoreArticle7 Aug 2026Computing in the Integers Modulo nImplementing arithmetic in Z_n: representative choice, reduction after each operation, inversion, and the cost of each primitive.3 min readRead MoreArticle7 Aug 2026Modular Exponentiation by Repeated SquaringSquare-and-multiply exponentiation, its cost, windowed variants, and the side-channel hazards of the naive form.3 min readRead MoreArticle7 Aug 2026Faster Integer Arithmetic: Karatsuba and BeyondKaratsuba multiplication and the divide-and-conquer family that reduces the exponent below two.3 min readRead MoreArticle7 Aug 2026Euclid's Algorithm for Integer GCDEuclid's algorithm for greatest common divisors, its correctness, and the Fibonacci worst case that bounds its iteration count.3 min readRead MoreArticle7 Aug 2026The Extended Euclidean AlgorithmComputing Bezout coefficients alongside the gcd, the recurrence for the coefficient sequences, and the size bounds that make it practical.3 min readRead MoreArticle7 Aug 2026Modular Inverses and Chinese RemainderingCombining modular inversion with Chinese remaindering to move computations between a composite modulus and its coprime factors.3 min readRead MoreArticle7 Aug 2026Speeding Up Algorithms via Modular ComputationThe modular method for exact computation: bounding the result, computing modulo several primes, and reconstructing.3 min readRead MoreArticle7 Aug 2026Rational ReconstructionRecovering a rational number from its residue modulo n, the uniqueness conditions, and the role of the extended Euclidean algorithm.3 min readRead MoreArticle7 Aug 2026Rational Reconstruction in Symbolic AlgebraApplying rational reconstruction inside computer algebra systems for exact linear solving, interpolation and gcd computation.3 min readRead MoreArticle7 Aug 2026Chebyshev's Theorem on the Density of PrimesChebyshev's elementary bounds on the prime counting function, the binomial coefficient argument, and what they establish short of the prime number theorem.3 min readRead MoreArticle7 Aug 2026Bertrand's PostulateBertrand's postulate that a prime always lies between n and 2n, its elementary proof, and its use in algorithm analysis.3 min readRead MoreArticle7 Aug 2026Mertens' TheoremMertens' theorems on sums and products over primes, and their role in estimating smoothness probabilities.3 min readRead MoreArticle7 Aug 2026The Sieve of EratosthenesThe classical sieve for enumerating primes, its complexity, segmented variants, and its role as a precomputation step.3 min readRead MoreArticle7 Aug 2026The Prime Number TheoremThe prime number theorem, its equivalent formulations, and the logarithmic integral as the superior approximation.3 min readRead MoreArticle7 Aug 2026The Error Term in the Prime Number TheoremHow closely li(x) approximates pi(x), the connection to zeta zeros, and what the Riemann hypothesis would give.3 min readRead MoreArticle7 Aug 2026Explicit Estimates for Prime CountingEffective, fully explicit bounds on the prime counting function and the nth prime, usable directly in algorithm analysis.3 min readRead MoreArticle7 Aug 2026Primes in Arithmetic ProgressionsDirichlet's theorem on primes in arithmetic progressions, equidistribution across residue classes, and computational consequences.3 min readRead MoreArticle7 Aug 2026Sophie Germain PrimesSophie Germain primes and safe primes, their use in discrete logarithm cryptography, and the conjectural nature of their density.3 min readRead MoreArticle7 Aug 2026Finite Probability DistributionsFinite sample spaces, probability distributions, events and the basic laws governing them.3 min readRead MoreArticle7 Aug 2026Conditional Probability and IndependenceConditional probability, Bayes' theorem, independence and the distinction between pairwise and mutual independence.3 min readRead MoreArticle7 Aug 2026Random VariablesRandom variables, their distributions, joint behaviour and independence.3 min readRead MoreArticle7 Aug 2026Expectation and VarianceExpectation, variance, their algebraic properties and their use in analysing randomised algorithms.3 min readRead MoreArticle7 Aug 2026Markov's and Chebyshev's InequalitiesMarkov's and Chebyshev's inequalities, their proofs, and how they bound deviation from the mean.3 min readRead MoreArticle7 Aug 2026The Birthday ParadoxThe birthday problem, the square-root threshold for collisions, and its algorithmic consequences.3 min readRead MoreArticle7 Aug 2026Hash Function FamiliesFamilies of hash functions, keyed selection, and the properties required of them in algorithm design.3 min readRead MoreArticle7 Aug 2026Pairwise Independence and Universal Hash FamiliesConstructing pairwise independent hash families over finite fields and why the weaker independence suffices.3 min readRead MoreArticle7 Aug 2026Hash TablesHash tables analysed with universal families: expected chain length, load factor and collision resolution.3 min readRead MoreArticle7 Aug 2026Message Authentication with Hash FunctionsUnconditionally secure message authentication from universal hash families, and how forgery probability is bounded.3 min readRead MoreArticle7 Aug 2026Statistical DistanceStatistical distance between distributions, its properties, and its use in proving that a sampler is close to uniform.3 min readRead MoreArticle7 Aug 2026Measures of Randomness and the Leftover Hash LemmaMin-entropy, randomness extraction, and the leftover hash lemma that converts weak randomness into near-uniform bits.3 min readRead MoreArticle7 Aug 2026Infinite Discrete Probability DistributionsCountably infinite sample spaces, convergence conditions, and the geometric distribution arising from unbounded loops.3 min readRead MoreArticle7 Aug 2026Probabilistic Algorithms: FoundationsThe model of randomised computation, Las Vegas and Monte Carlo algorithms, and what a probabilistic guarantee means.3 min readRead MoreArticle7 Aug 2026Reducing the Error ProbabilityAmplifying the success probability of a randomised algorithm by independent repetition, for one-sided and two-sided error.3 min readRead MoreArticle7 Aug 2026Strict Polynomial TimeStrict versus expected polynomial time, and how a Las Vegas algorithm is converted into a bounded-time one.3 min readRead MoreArticle7 Aug 2026Language Recognition and Complexity ClassesRandomised complexity classes RP, co-RP, BPP and ZPP, and where primality testing sits among them.3 min readRead MoreArticle7 Aug 2026Approximating Functions by Random SamplingEstimating a quantity by sampling, the sample size required, and the resulting confidence guarantee.3 min readRead MoreArticle7 Aug 2026Flipping a Coin Until a Head AppearsThe geometric waiting time, its expectation and tail, and its role as the model for repeat-until-success algorithms.3 min readRead MoreArticle7 Aug 2026Generating a Random Number from a Given IntervalSampling uniformly from an arbitrary range using a source of random bits, and controlling the resulting bias.3 min readRead MoreArticle7 Aug 2026Generating a Random PrimeGenerating a random prime by repeated candidate testing, the expected number of trials, and the sieving optimisation.3 min readRead MoreArticle7 Aug 2026Generating a Random k-Bit PrimeGenerating a prime of exactly k bits, the density in that range, and the constraints imposed by cryptographic use.3 min readRead MoreArticle7 Aug 2026Generating a Random Non-Increasing SequenceSampling a random non-increasing sequence in a bounded range, and its role as a subroutine in generating factored numbers.3 min readRead MoreArticle7 Aug 2026Generating a Random Factored NumberProducing a uniformly random integer together with its complete factorisation, in polynomial time, without factoring.3 min readRead MoreArticle7 Aug 2026The RSA CryptosystemRSA key generation, encryption, decryption and correctness, together with the assumptions its security depends on.3 min readRead MoreArticle7 Aug 2026Abelian Groups: Definitions, Properties and ExamplesAbelian groups: axioms, standard examples, and why the commutative case is sufficient for computational number theory.3 min readRead MoreArticle7 Aug 2026The Order of a Group ElementThe order of an element, its relationship to the group order, and the computational cost of determining it.3 min readRead MoreArticle7 Aug 2026SubgroupsSubgroups, the subgroup test, generated subgroups, and the subgroup lattice of a finite group.3 min readRead MoreArticle7 Aug 2026Cosets and Lagrange's TheoremCosets as a partition of a group, Lagrange's theorem, and its consequences for element orders.3 min readRead MoreArticle7 Aug 2026Quotient GroupsConstructing the quotient group from a subgroup, well-definedness of the induced operation, and the standard examples.3 min readRead MoreArticle7 Aug 2026Group Homomorphisms and IsomorphismsStructure-preserving maps between groups, isomorphisms, and what it means for two groups to be the same.3 min readRead MoreArticle7 Aug 2026Kernels, Images and the Isomorphism TheoremsThe kernel and image of a homomorphism, and the first isomorphism theorem relating them to a quotient.3 min readRead MoreArticle7 Aug 2026Cyclic GroupsCyclic groups, their generators, subgroup structure, and the criterion for the units modulo n to be cyclic.3 min readRead MoreArticle7 Aug 2026The Structure of Finite Abelian GroupsThe structure theorem decomposing every finite abelian group into cyclic factors, and its computational consequences.3 min readRead MoreArticle7 Aug 2026Rings: Definitions, Properties and ExamplesCommutative rings with unity: axioms, units, and the standard examples used throughout the subject.3 min readRead MoreArticle7 Aug 2026Zero Divisors and Integral DomainsZero divisors, integral domains, and why the absence of zero divisors is what makes cancellation and root counting work.3 min readRead MoreArticle7 Aug 2026SubringsSubrings, the subring test, and the distinction between subrings and ideals.3 min readRead MoreArticle7 Aug 2026Polynomials versus Polynomial FunctionsThe distinction between a formal polynomial and the function it induces, and why the two differ over finite rings.3 min readRead MoreArticle7 Aug 2026Basic Properties of Polynomial RingsDegree, leading coefficients, the ring structure of R[X], and when it is an integral domain.3 min readRead MoreArticle7 Aug 2026Formal Derivatives of PolynomialsThe formal derivative as an algebraic operation, its rules, and its use in detecting repeated factors.3 min readRead MoreArticle7 Aug 2026Multi-Variate PolynomialsPolynomials in several variables, total and partial degree, and the structural differences from the univariate case.3 min readRead MoreArticle7 Aug 2026Ideals and Quotient RingsIdeals, principal ideal domains, quotient rings, and the condition under which a quotient is a field.3 min readRead MoreArticle7 Aug 2026Ring Homomorphisms and IsomorphismsRing homomorphisms, kernels as ideals, and the first isomorphism theorem for rings.3 min readRead MoreArticle7 Aug 2026Trial Division and Basic Primality TestingTrial division as a primality test and as a filter, its exponential cost, and the role it still plays in practice.3 min readRead MoreArticle7 Aug 2026The Structure of the Group of Units Modulo nThe structure of Z_n* as a product of cyclic groups, the Carmichael function, and why prime moduli behave differently.3 min readRead MoreArticle7 Aug 2026The Fermat Test and Carmichael NumbersThe Fermat primality test, pseudoprimes, and the Carmichael numbers that defeat it for every base.3 min readRead MoreArticle7 Aug 2026The Miller-Rabin Primality TestThe Miller-Rabin test, the witness structure, the one-quarter bound, and why it is the practical standard.3 min readRead MoreArticle7 Aug 2026Generating a Random Prime Between 2 and MGenerating a uniform random prime below a bound, the analysis of the retry loop, and the resulting output distribution.3 min readRead MoreArticle7 Aug 2026Trial Division up to a Small BoundChoosing the trial division bound ahead of a probabilistic test, and the cost balance that determines it.3 min readRead MoreArticle7 Aug 2026Generating a Random k-Bit Prime with Miller-RabinAssembling bit-length constraint, trial division filtering and Miller-Rabin into a complete prime generator.3 min readRead MoreArticle7 Aug 2026Perfect Power Testing and Prime Power FactoringDetecting whether an integer is a perfect power, extracting the root, and why this precedes general factoring.3 min readRead MoreArticle7 Aug 2026Factoring and Computing Euler's Phi FunctionThe polynomial-time equivalence between factoring a modulus, computing phi, and recovering an RSA private exponent.3 min readRead MoreArticle7 Aug 2026Deterministic Primality Testing: The Basic IdeaThe polynomial identity underlying AKS primality testing and the obstacle that makes it non-trivial to exploit.3 min readRead MoreArticle7 Aug 2026The AKS Algorithm and Its AnalysisThe AKS algorithm in full, its correctness argument, complexity, and why it is not used in practice.3 min readRead MoreArticle7 Aug 2026Finding a Generator of the Group of Units Modulo pLocating a generator of Z_p*, the test based on the factorisation of p-1, and the density of generators.3 min readRead MoreArticle7 Aug 2026Brute-Force Discrete Logarithm SearchExhaustive search for discrete logarithms, its cost, and its role as the baseline against which other methods are measured.3 min readRead MoreArticle7 Aug 2026The Baby Step/Giant Step MethodShanks' baby step giant step algorithm, its square-root running time, and the time-memory trade-off it embodies.3 min readRead MoreArticle7 Aug 2026Discrete Logarithms in Groups of Prime Power OrderSolving discrete logarithms in a group of prime power order by digit-by-digit lifting.3 min readRead MoreArticle7 Aug 2026Discrete Logarithms in the Full Group Modulo pThe Pohlig-Hellman reduction combining prime power subproblems by the Chinese remainder theorem.3 min readRead MoreArticle7 Aug 2026The Diffie-Hellman Key Establishment ProtocolDiffie-Hellman key agreement, the assumptions it rests on, and the authentication gap that makes it vulnerable alone.3 min readRead MoreArticle7 Aug 2026Smooth NumbersSmooth numbers, their density, and why they are the raw material of subexponential factoring and index calculus.3 min readRead MoreArticle7 Aug 2026Subexponential Discrete Logarithm AlgorithmsIndex calculus for discrete logarithms in Z_p*, its two phases, and the precomputation asymmetry it creates.3 min readRead MoreArticle7 Aug 2026Subexponential Integer FactoringFactoring by congruences of squares, the relation collection and linear algebra phases, and the resulting subexponential cost.3 min readRead MoreArticle7 Aug 2026Better Smoothness Density EstimatesRefined estimates for smooth number density and how they determine optimal sieve parameters.3 min readRead MoreArticle7 Aug 2026The Quadratic Sieve AlgorithmThe quadratic sieve: candidate generation near the square root, sieving for smoothness, and its practical range.3 min readRead MoreArticle7 Aug 2026The Number Field Sieve and Factoring RecordsThe number field sieve, the current state of factoring records, and how key size recommendations follow from them.3 min readRead MoreArticle7 Aug 2026Quadratic ResiduesQuadratic residues modulo a prime, the exact split into residues and non-residues, and the group-theoretic reason for it.3 min readRead MoreArticle7 Aug 2026The Legendre SymbolThe Legendre symbol as a multiplicative character, its evaluation by Euler's criterion, and the supplementary laws.3 min readRead MoreArticle7 Aug 2026The Law of Quadratic ReciprocityThe law of quadratic reciprocity, its statement, and why it makes symbol evaluation efficient.3 min readRead MoreArticle7 Aug 2026The Jacobi SymbolThe Jacobi symbol as the multiplicative extension of the Legendre symbol to odd composite moduli, and what it does and does not tell you.3 min readRead MoreArticle7 Aug 2026Computing the Jacobi SymbolThe Euclid-style algorithm for evaluating a Jacobi symbol in quadratic time without factoring either argument.3 min readRead MoreArticle7 Aug 2026Testing Quadratic Residuosity: Prime ModulusDeciding quadratic residuosity modulo a prime, and why the problem is easy in this case.3 min readRead MoreArticle7 Aug 2026Testing Quadratic Residuosity: Prime Power and Composite ModulusQuadratic residuosity modulo prime powers and composites, the reduction by Chinese remaindering, and where the hardness enters.3 min readRead MoreArticle7 Aug 2026Computing Modular Square Roots: Prime ModulusExtracting square roots modulo a prime, the easy case for p congruent to 3 mod 4, and the randomised algorithm in general.3 min readRead MoreArticle7 Aug 2026Computing Modular Square Roots: Prime Power ModulusLifting a square root from a prime to a prime power by Hensel's method, and the special handling powers of two require.3 min readRead MoreArticle7 Aug 2026Computing Modular Square Roots: Composite ModulusSquare roots modulo a composite, the four roots for a semiprime, and the equivalence with factoring.3 min readRead MoreArticle7 Aug 2026The Quadratic Residuosity AssumptionThe quadratic residuosity assumption, its use in probabilistic encryption, and its relationship to factoring.3 min readRead MoreArticle7 Aug 2026Modules: Definitions, Properties and ExamplesModules over a commutative ring, the generalisation of vector spaces, and what changes when scalars need not be invertible.3 min readRead MoreArticle7 Aug 2026Submodules and Quotient ModulesSubmodules, quotient modules, and the correspondence between submodules of a quotient and those of the original.3 min readRead MoreArticle7 Aug 2026Module Homomorphisms and IsomorphismsModule homomorphisms, kernels and images, and the isomorphism theorems in their module form.3 min readRead MoreArticle7 Aug 2026Linear Independence and BasesLinear independence, spanning sets and bases, and the conditions under which a basis exists.3 min readRead MoreArticle7 Aug 2026Vector Spaces and DimensionVector spaces over a field, the well-definedness of dimension, and the rank-nullity relation.3 min readRead MoreArticle7 Aug 2026Matrices: Basic Definitions and PropertiesMatrices over a ring, their arithmetic, and the properties that survive when the base ring is not a field.3 min readRead MoreArticle7 Aug 2026Matrices and Linear MapsThe correspondence between matrices and linear maps, change of basis, and why the correspondence depends on a choice.3 min readRead MoreArticle7 Aug 2026The Inverse of a MatrixMatrix inversion over a field, its computation by elimination, and why explicit inversion is usually the wrong operation.3 min readRead MoreArticle7 Aug 2026Gaussian EliminationGaussian elimination over a field, its complexity, pivoting, and its role as the bottleneck in sieve algorithms.3 min readRead MoreArticle7 Aug 2026Computing Rank, Kernel and ImageExtracting rank, kernel basis and image basis from an echelon form, and the applications to polynomial factorisation.3 min readRead MoreArticle7 Aug 2026Solving Systems of Linear EquationsSolving linear systems over a field: consistency, the structure of the solution set, and modular methods for exact rational answers.3 min readRead MoreArticle7 Aug 2026Algebras over a RingAlgebras as rings carrying a compatible module structure, and the examples that matter computationally.3 min readRead MoreArticle7 Aug 2026The Field of Fractions of an Integral DomainConstructing the field of fractions of an integral domain, its universal property, and the standard examples.3 min readRead MoreArticle7 Aug 2026Unique Factorization of PolynomialsUnique factorisation in polynomial rings over a field, and the extension to polynomial rings over a UFD.3 min readRead MoreArticle7 Aug 2026Irreducible PolynomialsRecognising irreducible polynomials, the standard criteria, and counting them over a finite field.3 min readRead MoreArticle7 Aug 2026Polynomial CongruencesCongruences of polynomials modulo a fixed polynomial, and the parallel with integer congruences.3 min readRead MoreArticle7 Aug 2026Polynomial Quotient AlgebrasThe structure of F[X]/(f), its basis, arithmetic, and the decomposition when f is reducible.3 min readRead MoreArticle7 Aug 2026General Properties of Extension FieldsField extensions, degree, algebraic elements and minimal polynomials.3 min readRead MoreArticle7 Aug 2026Formal Power SeriesFormal power series, their arithmetic, invertibility criterion, and use as generating functions.3 min readRead MoreArticle7 Aug 2026Formal Laurent SeriesFormal Laurent series, the field of fractions of the power series ring, and the valuation structure.3 min readRead MoreArticle7 Aug 2026Reversed Formal Laurent SeriesReversed formal Laurent series, the valuation by degree, and their role in rational function reconstruction.3 min readRead MoreArticle7 Aug 2026Unique Factorization DomainsUnique factorisation domains, the distinction between irreducible and prime, and the standard examples and counterexamples.3 min readRead MoreArticle7 Aug 2026Unique Factorization in Euclidean and Principal Ideal DomainsWhy Euclidean domains are principal ideal domains and why principal ideal domains have unique factorisation.3 min readRead MoreArticle7 Aug 2026Basic Polynomial ArithmeticRepresentation of polynomials, addition, multiplication and division costs, and the dense versus sparse choice.3 min readRead MoreArticle7 Aug 2026Computing Minimal Polynomials in Quotient AlgebrasFinding the minimal polynomial of an element of a polynomial quotient algebra by linear algebra on its powers.3 min readRead MoreArticle7 Aug 2026Euclid's Algorithm for PolynomialsThe Euclidean algorithm for polynomial gcds, its degree-based termination, and coefficient growth over the rationals.3 min readRead MoreArticle7 Aug 2026Polynomial Modular InversesInverting a polynomial modulo another using the extended Euclidean algorithm, and the finite field application.3 min readRead MoreArticle7 Aug 2026Chinese Remaindering and Polynomial InterpolationThe Chinese remainder theorem for polynomials, and Lagrange interpolation as its special case.3 min readRead MoreArticle7 Aug 2026Mutual Independence and Secret SharingShamir's threshold secret sharing, its information-theoretic security, and the independence property that underlies it.3 min readRead MoreArticle7 Aug 2026Speeding Up Polynomial Algorithms via Modular ComputationApplying evaluation homomorphisms and modular reduction to control coefficient and degree growth in polynomial computation.3 min readRead MoreArticle7 Aug 2026Rational Function ReconstructionRecovering a rational function from a residue modulo a polynomial, with degree bounds replacing size bounds.3 min readRead MoreArticle7 Aug 2026Error-Correcting Codes and Algebraic DecodingReed-Solomon codes, their distance property, and decoding by rational function reconstruction.3 min readRead MoreArticle7 Aug 2026Rational Function Reconstruction in Symbolic AlgebraUsing rational function reconstruction inside computer algebra for exact computation over function fields.3 min readRead MoreArticle7 Aug 2026Faster Polynomial ArithmeticSubquadratic polynomial multiplication by Karatsuba and FFT methods, and the crossover behaviour.3 min readRead MoreArticle7 Aug 2026Linearly Generated SequencesSequences satisfying linear recurrences, their minimal polynomials, and the equivalence with rational generating functions.3 min readRead MoreArticle7 Aug 2026Computing Minimal Polynomials of SequencesThe Berlekamp-Massey algorithm and its Euclidean equivalent for finding the shortest linear recurrence.3 min readRead MoreArticle7 Aug 2026Solving Sparse Linear SystemsIterative methods for large sparse systems over finite fields, and their role as the bottleneck of sieve algorithms.3 min readRead MoreArticle7 Aug 2026The Algebra of Linear TransformationsThe endomorphism algebra of a vector space, minimal and characteristic polynomials, and the module view of a linear operator.3 min readRead MoreArticle7 Aug 2026Finite Fields: PreliminariesThe characteristic of a finite field, why its order is a prime power, and the prime subfield.3 min readRead MoreArticle7 Aug 2026The Existence of Finite FieldsConstruction of a field of any prime power order, and the proof that one exists for every such order.3 min readRead MoreArticle7 Aug 2026Subfield Structure and Uniqueness of Finite FieldsThe subfield lattice of a finite field, its correspondence with divisors, and the uniqueness of each subfield.3 min readRead MoreArticle7 Aug 2026Conjugates, Norms and TracesConjugates of a finite field element, the norm and trace maps, and their surjectivity onto the subfield.3 min readRead MoreArticle7 Aug 2026The Frobenius MapThe Frobenius endomorphism, its fixed field, its order, and the Galois structure of finite field extensions.3 min readRead MoreArticle7 Aug 2026Testing and Constructing Irreducible PolynomialsTesting irreducibility over a finite field and constructing irreducible polynomials of prescribed degree.3 min readRead MoreArticle7 Aug 2026Computing Minimal Polynomials over Finite FieldsComputing the minimal polynomial of a finite field element using conjugates or linear algebra.3 min readRead MoreArticle7 Aug 2026Distinct Degree FactorizationSeparating the irreducible factors of a polynomial by degree using gcds with Frobenius powers.3 min readRead MoreArticle7 Aug 2026Equal Degree FactorizationSplitting a product of same-degree irreducible factors by random splitting, and the success probability.4 min readRead MoreArticle7 Aug 2026Analysis of the Cantor-Zassenhaus AlgorithmThe complete Cantor-Zassenhaus factorisation algorithm, its three stages, and its overall cost.3 min readRead MoreArticle7 Aug 2026Square-Free Decomposition of PolynomialsRemoving repeated factors using gcds with the derivative, and the characteristic p complication.3 min readRead MoreArticle7 Aug 2026Berlekamp's Factorization AlgorithmBerlekamp's algorithm: the Berlekamp subalgebra, the kernel computation, and splitting by gcds.3 min readRead MoreArticle7 Aug 2026Analysis of Berlekamp's AlgorithmCost analysis of Berlekamp's algorithm and the variants that improve its dependence on field size.3 min readRead MoreArticle7 Aug 2026Deterministic Polynomial Factorization AlgorithmsDeterministic factorisation over finite fields, the obstacles, and what is known conditionally.3 min readRead MoreArticle7 Aug 2026Faster Square-Free DecompositionYun's algorithm and other improvements to squarefree decomposition, and their cost advantages.3 min readRead MoreArticle7 Aug 2026Computational Number Theory: Tools and LibrariesSoftware for computational number theory and algebra, what each is suited to, and how to choose.3 min readRead MoreArticle7 Aug 2026Arbitrary Precision Arithmetic in PracticeImplementation concerns for multiprecision arithmetic: memory management, algorithm dispatch, and constant-time requirements.3 min readRead MoreArticle7 Aug 2026Parameter Sizes, Records and Live ReferencesWhere to obtain current key size recommendations, factoring records and post-quantum guidance, and why they are not reproduced here.3 min readRead MoreArticle7 Aug 2026Computational Algebraic Number Theory: Field OverviewWhat computational algebraic number theory actually computes, why the problems are hard, and how the subject's algorithms fit together.4 min readRead MoreArticle7 Aug 2026Algorithm Notation and Complexity ConventionsThe notation, cost model and complexity conventions used throughout this collection, including the sub-exponential L-function.3 min readRead MoreArticle7 Aug 2026Learning Pathways Through Computational Number TheoryFour routes through this 182-page collection, chosen by what you need to build rather than by chapter order.2 min readRead MoreArticle7 Aug 2026The Four Core Computational Tasks of Number FieldsThe four fundamental computational problems for a number field, their dependencies, and what counts as a complete answer to each.4 min readRead MoreArticle7 Aug 2026Multiprecision Integer RepresentationHow arbitrary-precision integers are stored, why the base is chosen to match the machine word, and the consequences for every algorithm above.2 min readRead MoreArticle7 Aug 2026Multiprecision Addition and SubtractionCarry and borrow propagation, sign handling, and why addition sets the baseline cost against which every other operation is measured.2 min readRead MoreArticle7 Aug 2026Schoolbook and Karatsuba MultiplicationQuadratic schoolbook multiplication, the Karatsuba three-multiplication identity, and where the crossover between them sits.3 min readRead MoreArticle7 Aug 2026Asymptotic Cost of Integer MultiplicationThe M(n) abstraction, the hierarchy of multiplication algorithms, and why downstream bounds are quoted in terms of M(n) rather than fixed exponents.2 min readRead MoreArticle7 Aug 2026Multiprecision Division and RemainderKnuth's division algorithm, the normalisation step that makes quotient digit estimation reliable, and why division carries a larger constant than multiplication.2 min readRead MoreArticle7 Aug 2026Modular Arithmetic and Montgomery ReductionModular reduction strategies, Montgomery representation, and how trading division for multiplication accelerates every exponentiation.2 min readRead MoreArticle7 Aug 2026Binary Powering and Exponentiation ChainsSquare-and-multiply exponentiation, left-to-right and right-to-left variants, windowing, and why exponentiation cost drives primality testing.2 min readRead MoreArticle7 Aug 2026Integer Square Root and Perfect Power DetectionNewton iteration for integer square roots, exact perfect power detection, and why these cheap tests belong at the front of every factoring routine.2 min readRead MoreArticle7 Aug 2026The Euclidean Algorithm: Classical and Binary VariantsThe classical Euclidean algorithm, the binary variant that replaces division with shifts, and how to choose between them.2 min readRead MoreArticle7 Aug 2026Lehmer's Accelerated GCD ComputationLehmer's method: running many GCD steps on single-precision leading digits before touching the full multiprecision operands.2 min readRead MoreArticle7 Aug 2026The Extended Euclidean Algorithm and Bezout CoefficientsComputing Bezout coefficients alongside the GCD, modular inversion as its principal application, and controlling coefficient growth.2 min readRead MoreArticle7 Aug 2026Chinese Remainder Theorem AlgorithmsReconstructing an integer from residues, Garner's incremental method, and CRT as a strategy for controlling coefficient growth.2 min readRead MoreArticle7 Aug 2026Continued Fraction Expansion of Real NumbersContinued fractions, convergents as best rational approximations, and the periodic expansion of quadratic irrationals.2 min readRead MoreArticle7 Aug 2026Structure of the Unit Group Modulo nThe structure of the multiplicative group of integers modulo n, its decomposition by CRT, and computing element orders.2 min readRead MoreArticle7 Aug 2026Legendre, Jacobi and Kronecker Symbol ComputationThe three quadratic symbols, their differing meanings, and the reciprocity-based algorithm that computes them without factoring.2 min readRead MoreArticle7 Aug 2026Square Roots Modulo a Prime: the Shanks-Tonelli AlgorithmExtracting square roots modulo a prime: the easy congruence classes, the general Shanks-Tonelli algorithm, and lifting to prime powers.2 min readRead MoreArticle7 Aug 2026Solving Quadratic CongruencesReducing a general quadratic congruence to a square root extraction, and handling the degenerate cases the reduction assumes away.2 min readRead MoreArticle7 Aug 2026Modular Inversion and Simultaneous InversionComputing modular inverses, and Montgomery's trick for inverting many elements at the cost of one inversion plus multiplications.2 min readRead MoreArticle7 Aug 2026Finite Field Element RepresentationRepresenting elements of a finite field as polynomials modulo an irreducible, choosing the modulus, and the trade-offs against logarithmic and normal bases.2 min readRead MoreArticle7 Aug 2026Finite Field Multiplication and InversionMultiplication with reduction, inversion by extended Euclid or by exponentiation, and the Itoh-Tsujii method for extension fields.2 min readRead MoreArticle7 Aug 2026Finding Primitive Roots and GeneratorsLocating a generator of a cyclic group, why the search is easy but verification requires a factorisation, and the deterministic gap.2 min readRead MoreArticle7 Aug 2026Root Finding over Finite FieldsFinding roots of a polynomial in a finite field by GCD with the Frobenius polynomial followed by probabilistic splitting.2 min readRead MoreArticle7 Aug 2026Finite Field Arithmetic in PracticeSelecting representations and algorithms for real finite field workloads, with the operation mixes that arise in factoring, decomposition and point counting.2 min readRead MoreArticle7 Aug 2026Matrix Representation and Cost ModelDense and sparse matrix representations, the cost model for exact linear algebra, and why coefficient growth rather than operation count usually decides performance.2 min readRead MoreArticle7 Aug 2026Gaussian Elimination and Linear SystemsExact Gaussian elimination, pivoting for entry growth rather than stability, and the fraction-free Bareiss variant.2 min readRead MoreArticle7 Aug 2026Gaussian Elimination over Finite FieldsElimination over a finite field, structured methods for very large sparse systems, and why the linear algebra stage limits sieving methods.2 min readRead MoreArticle7 Aug 2026Determinant Computation StrategiesComputing exact determinants by fraction-free elimination, modular methods with Hadamard bounds, and how to choose between them.2 min readRead MoreArticle7 Aug 2026The Characteristic Polynomial and the Hessenberg MethodComputing the characteristic polynomial via Hessenberg reduction, and its role in producing minimal polynomials of algebraic numbers.2 min readRead MoreArticle7 Aug 2026Kernel and Image of a General MatrixComputing kernel and image bases for rectangular matrices over a field, and why the integer analogue is a different problem.2 min readRead MoreArticle7 Aug 2026Inverse Image and Supplementation of SubspacesSolving for preimages of a subspace, extending an independent set to a basis, and the linear algebra primitives these support.2 min readRead MoreArticle7 Aug 2026Operations on Subspaces and ModulesSum, intersection and quotient of subspaces, and how the analogous operations on modules over the integers require normal forms.2 min readRead MoreArticle7 Aug 2026Z-Modules and Integer Matrix ProblemsFinitely generated abelian groups as integer matrix problems, and the two normal forms that answer the two basic questions about them.2 min readRead MoreArticle7 Aug 2026The Hermite Normal Form AlgorithmThe Hermite normal form, the classical column-reduction algorithm, and the modular variant that bounds entry growth.2 min readRead MoreArticle7 Aug 2026Coefficient Explosion in Hermite Normal Form ComputationWhy intermediate entries in normal form computation grow so violently, how to recognise it, and the three standard mitigations.2 min readRead MoreArticle7 Aug 2026Applications of the Hermite Normal FormSolving integer linear systems, computing indices and intersections, and testing module membership using the Hermite normal form.2 min readRead MoreArticle7 Aug 2026The Smith Normal Form AlgorithmThe Smith normal form, its computation by alternating row and column reduction, and the invariant factors it exposes.2 min readRead MoreArticle7 Aug 2026Recovering Abelian Group Structure from a Relation MatrixRecovering the structure and explicit generators of a finite abelian group from a matrix of relations among a generating set.2 min readRead MoreArticle7 Aug 2026LLL-Based Hermite Normal Form ComputationUsing lattice reduction to control entry growth during Hermite normal form computation, and when this beats the modular approach.2 min readRead MoreArticle7 Aug 2026Lattice Definitions and Quadratic FormsLattices as discrete subgroups, the Gram matrix, and the correspondence between lattices with a basis and positive definite quadratic forms.2 min readRead MoreArticle7 Aug 2026The Gram-Schmidt Orthogonalisation ProcedureGram-Schmidt orthogonalisation, the coefficients that drive lattice reduction, and why the orthogonal vectors themselves are not lattice vectors.2 min readRead MoreArticle7 Aug 2026Lattice Determinant and the Hadamard BoundThe lattice determinant as a basis-independent invariant, the Hadamard inequality, and using the orthogonality defect to measure basis quality.2 min readRead MoreArticle7 Aug 2026The LLL Lattice Basis Reduction AlgorithmThe LLL algorithm: size reduction interleaved with swaps under the Lovasz condition, and the guarantees it provides in polynomial time.3 min readRead MoreArticle7 Aug 2026The LLL Lattice Basis Reduction AlgorithmThe LLL algorithm: size reduction interleaved with swaps under the Lovasz condition, and the guarantees it provides in polynomial time.3 min readRead MoreArticle7 Aug 2026LLL Reduction Quality and Proof SketchWhy LLL terminates in polynomial time and what its output guarantees, via the potential function and the Lovasz condition.2 min readRead MoreArticle7 Aug 2026LLL with Deep InsertionsAllowing a vector to move further than one position, the quality gain, and the loss of the polynomial time guarantee.2 min readRead MoreArticle7 Aug 2026Integral LLL: Avoiding Floating PointRunning LLL entirely in integer arithmetic using scaled Gram-Schmidt quantities, and why this matters for exact downstream computation.2 min readRead MoreArticle7 Aug 2026LLL for Linearly Dependent Generating SetsExtending LLL to generating sets that are not independent, and using the resulting zero vectors to extract relations.2 min readRead MoreArticle7 Aug 2026Integer Kernel and Image via LLLComputing a reduced basis of the integer kernel and image of a matrix, and why this is not the same as clearing denominators from a rational kernel.2 min readRead MoreArticle7 Aug 2026Detecting Algebraic and Linear Dependence with LLLRecovering exact integer relations from numerical approximations using LLL, and the precision requirements that make the method reliable.2 min readRead MoreArticle7 Aug 2026Finding Short Vectors in LatticesEnumeration and reduction-based methods for finding short lattice vectors, and where the exact shortest vector is genuinely needed.2 min readRead MoreArticle7 Aug 2026Polynomial Representation and StorageDense and sparse polynomial representations, coefficient domains, and the normalisation invariants every implementation must maintain.2 min readRead MoreArticle7 Aug 2026Polynomial Multiplication StrategiesSchoolbook, Karatsuba and evaluation-interpolation methods for polynomial multiplication, and where the crossovers lie.2 min readRead MoreArticle7 Aug 2026Polynomial Division with RemainderEuclidean division of polynomials, pseudo-division over a ring without inverses, and the coefficient growth pseudo-division introduces.2 min readRead MoreArticle7 Aug 2026The Polynomial Euclidean Algorithm over a FieldThe Euclidean and extended Euclidean algorithms for polynomials over a field, and their role in inversion and interpolation.2 min readRead MoreArticle7 Aug 2026Unique Factorisation Domains, Content and Primitive PartsContent and primitive part, Gauss's lemma, and why factoring over the rationals reduces to factoring over the integers.2 min readRead MoreArticle7 Aug 2026Polynomial GCD over a Unique Factorisation DomainComputing polynomial GCDs over the integers, the growth problem in remainder sequences, and the modular approach that sidesteps it.2 min readRead MoreArticle7 Aug 2026The Sub-Resultant GCD AlgorithmThe sub-resultant remainder sequence: predicting the divisible factor at each step to keep coefficients near minimal without content computation.2 min readRead MoreArticle7 Aug 2026Resultants and DiscriminantsThe resultant as a criterion for common roots, the discriminant as a test for repeated roots, and how both are computed in practice.2 min readRead MoreArticle7 Aug 2026Polynomial Factorisation: Overall StrategyThe three-stage pipeline used to factor polynomials over finite fields and the integers, and why the stages are ordered as they are.2 min readRead MoreArticle7 Aug 2026Squarefree Factorisation of PolynomialsSeparating repeated factors using the derivative, and the modification required in positive characteristic.2 min readRead MoreArticle7 Aug 2026Distinct Degree FactorisationSeparating irreducible factors by degree using GCDs against Frobenius powers, and the early-abort strategies that make it fast.2 min readRead MoreArticle7 Aug 2026Cantor-Zassenhaus Equal Degree SplittingSplitting a product of irreducibles of equal degree by random elements, the probability analysis, and the characteristic two variant.2 min readRead MoreArticle7 Aug 2026The Berlekamp Factorisation AlgorithmBerlekamp's linear algebra approach to factoring over a finite field, the Berlekamp subalgebra, and when it outperforms the GCD pipeline.2 min readRead MoreArticle7 Aug 2026Mignotte Bounds on Polynomial FactorsBounding the coefficients of any factor of an integer polynomial, and why such a bound makes modular and lifting methods complete algorithms.2 min readRead MoreArticle7 Aug 2026Hensel Lifting for Polynomial FactorsLifting a factorisation modulo a prime to a factorisation modulo a prime power, the quadratic variant, and the coprimality condition.2 min readRead MoreArticle7 Aug 2026Factoring Polynomials over the IntegersThe modular-lift-recombine pipeline, the exponential recombination problem, and the LLL-based algorithm that makes factorisation polynomial time.2 min readRead MoreArticle7 Aug 2026Factoring Polynomials over Algebraic Number FieldsFactoring polynomials whose coefficients lie in a number field, by reduction to the rational case via norms.2 min readRead MoreArticle7 Aug 2026Root Finding over the Reals and Complex NumbersNumerical root finding for polynomials with exact coefficients, root isolation over the reals, and the precision required to be reliable.2 min readRead MoreArticle7 Aug 2026p-adic Root Finding and Newton PolygonsFinding roots in p-adic fields by lifting, and reading ramification structure off the Newton polygon of a polynomial.2 min readRead MoreArticle7 Aug 2026Algebraic Numbers and Minimal PolynomialsAlgebraic numbers, minimal polynomials, algebraic integers, and the computational tests that distinguish them.2 min readRead MoreArticle7 Aug 2026Number Fields: Definition and Basic PropertiesNumber fields as finite extensions of the rationals, their embeddings and signature, and the presentation on which all computation depends.2 min readRead MoreArticle7 Aug 2026The Standard Representation of Algebraic NumbersRepresenting field elements as coefficient vectors relative to a power basis or integral basis, with a common denominator.2 min readRead MoreArticle7 Aug 2026The Matrix (Regular) Representation of Algebraic NumbersRepresenting a field element by its multiplication matrix, and reading trace, norm and characteristic polynomial off that matrix.2 min readRead MoreArticle7 Aug 2026The Conjugate Vector RepresentationRepresenting a field element by its images under all embeddings, the analytic information this exposes, and the precision it demands.2 min readRead MoreArticle7 Aug 2026Trace, Norm and the Characteristic PolynomialTrace, norm and characteristic polynomial of a field element, their computation, and their use as invariants and cross-checks.2 min readRead MoreArticle7 Aug 2026Discriminants and Integral BasesThe discriminant of a basis, the field discriminant, and the index-squared relation that governs maximal order computation.2 min readRead MoreArticle7 Aug 2026The Polynomial Reduction AlgorithmFinding a small defining polynomial for a number field using lattice reduction on the maximal order, and why this pays for itself.2 min readRead MoreArticle7 Aug 2026The Subfield ProblemFinding the subfields of a number field, by lattice methods and by linear algebra over the complex numbers.2 min readRead MoreArticle7 Aug 2026Field Isomorphism and the Normal ClosureTesting whether two number fields are isomorphic, computing the isomorphisms, and constructing the normal closure.2 min readRead MoreArticle7 Aug 2026Orders in Number FieldsOrders as subrings that are full-rank lattices, the equation order, the maximal order, and the index that separates them.2 min readRead MoreArticle7 Aug 2026Ideals of the Maximal OrderIdeals and fractional ideals, unique factorisation into primes, and the group structure that makes the class group possible.2 min readRead MoreArticle7 Aug 2026Module Representation by Hermite Normal FormRepresenting ideals and modules as Hermite normal form matrices relative to an integral basis, with a common denominator.2 min readRead MoreArticle7 Aug 2026Ideal Representation by Two ElementsRepresenting an ideal by two generators, why two always suffice, and the trade-off against the canonical matrix form.2 min readRead MoreArticle7 Aug 2026Ideal Multiplication and DivisionMultiplying, inverting and dividing ideals as module operations, and controlling the growth these operations cause.2 min readRead MoreArticle7 Aug 2026Ideal Norm ComputationThe norm of an ideal as its index in the order, its multiplicativity, and its use as a size measure and consistency check.2 min readRead MoreArticle7 Aug 2026Prime Decomposition: Theory and RamificationHow rational primes factor in the maximal order, ramification indices and residue degrees, and the degree relation that constrains them.2 min readRead MoreArticle7 Aug 2026Prime Decomposition when p Does Not Divide the IndexDecomposing a prime that does not divide the index, by factoring the defining polynomial modulo that prime.2 min readRead MoreArticle7 Aug 2026Essential Discriminant DivisorsPrimes that divide the index for every choice of defining polynomial, why they exist, and what they force computationally.2 min readRead MoreArticle7 Aug 2026Valuations and UniformisersValuations at prime ideals, uniformising elements, and computing the exponent of a prime in an ideal factorisation.2 min readRead MoreArticle7 Aug 2026The Ideal Class GroupThe class group as the obstruction to unique factorisation, its finiteness, and what a complete computation must produce.2 min readRead MoreArticle7 Aug 2026The Dirichlet Unit Theorem, ComputationallyThe structure of the unit group, its rank from the signature, and what computing units actually requires.2 min readRead MoreArticle7 Aug 2026The Logarithmic Embedding and the Unit LatticeMapping units into a real vector space by logarithms of conjugate absolute values, and the lattice this produces.2 min readRead MoreArticle7 Aug 2026The Regulator: Definition and ComputationThe regulator as the covolume of the unit lattice, its computation, and the precision and verification it demands.2 min readRead MoreArticle7 Aug 2026Minkowski and Bach BoundsBounds on the norm of ideals needed to generate the class group, and why the conditional bound is what makes computation practical.2 min readRead MoreArticle7 Aug 2026Class Group and Unit Computation: the Computational ProblemWhy class group and unit computation are a single problem, what the combined algorithm produces, and how the results are verified.2 min readRead MoreArticle7 Aug 2026Quadratic Field Discriminants and Integral BasesDiscriminants and integral bases of quadratic fields, given by closed formulas with no computation required.2 min readRead MoreArticle7 Aug 2026Prime Decomposition in Quadratic FieldsHow rational primes split, remain inert or ramify in a quadratic field, decided entirely by the Kronecker symbol.2 min readRead MoreArticle7 Aug 2026Binary Quadratic Forms and the Ideal CorrespondenceThe dictionary between binary quadratic forms and ideals of a quadratic order, and why the form language is computationally preferable.2 min readRead MoreArticle7 Aug 2026Reduction of Positive Definite Binary FormsReducing a positive definite binary quadratic form to the unique reduced form in its class, and the resulting class number algorithm.2 min readRead MoreArticle7 Aug 2026Composition of Binary Quadratic FormsGauss composition as the group law on form classes, its practical computation, and reduction between compositions.2 min readRead MoreArticle7 Aug 2026Imaginary Quadratic Class Numbers by Counting Reduced FormsComputing class numbers and group structure for imaginary quadratic fields by enumerating reduced forms, and where the method runs out.2 min readRead MoreArticle7 Aug 2026Class Numbers from Analytic Class Number FormulasUsing L-functions and the analytic class number formula to compute or verify class numbers and regulators.2 min readRead MoreArticle7 Aug 2026Reduction of Indefinite Forms and the Cycle StructureReduction of indefinite binary forms, the cycle of reduced forms in each class, and how the cycle encodes the regulator.2 min readRead MoreArticle7 Aug 2026The Fundamental Unit of a Real Quadratic FieldComputing the fundamental unit of a real quadratic field by continued fractions, its connection to the Pell equation, and its size.2 min readRead MoreArticle7 Aug 2026Sub-exponential Class Group Computation for Quadratic FieldsSub-exponential class group and regulator computation for quadratic fields by relation collection over a factor base.2 min readRead MoreArticle7 Aug 2026Computing the Structure of Residue RingsThe structure of the unit group of a residue ring of the maximal order, and its computation by decomposition and lifting.2 min readRead MoreArticle7 Aug 2026The Maximal Order ProblemThe maximal order problem, its reduction to a prime-by-prime question, and the factoring obstruction that limits it.2 min readRead MoreArticle7 Aug 2026The Pohst-Zassenhaus TheoremThe theorem underlying maximal order computation: enlarging an order by the ring of multipliers of its radical.2 min readRead MoreArticle7 Aug 2026The Dedekind Criterion for p-MaximalityA cheap modular test deciding whether an order is maximal at a given prime, without computing the maximal order.2 min readRead MoreArticle7 Aug 2026The Round 2 Maximal Order AlgorithmThe Round 2 algorithm: computing the maximal order prime by prime via radicals and rings of multipliers.2 min readRead MoreArticle7 Aug 2026Radical Computation and the Ring of MultipliersComputing the p-radical of an order as a kernel, and the ring of multipliers that enlarges the order.2 min readRead MoreArticle7 Aug 2026Newton Polygon Methods for Prime DecompositionUsing Newton polygons to decompose primes locally, handling the cases where factoring modulo p is insufficient.2 min readRead MoreArticle7 Aug 2026Splitting Separable Algebras over Finite FieldsDecomposing a finite-dimensional commutative algebra over a finite field into its simple components, generalising polynomial factorisation.2 min readRead MoreArticle7 Aug 2026The Buchmann-Lenstra Prime Decomposition MethodDecomposing any prime, including those dividing the index, by splitting the algebra of the order modulo that prime.2 min readRead MoreArticle7 Aug 2026The Galois Group Computation ProblemWhat computing a Galois group means, why the splitting field cannot be constructed, and the transitive group classification that makes the problem finite.2 min readRead MoreArticle7 Aug 2026The Resolvent Method for Galois GroupsConstructing resolvent polynomials whose factorisation distinguishes candidate Galois groups, and the practical issues in using them.2 min readRead MoreArticle7 Aug 2026Galois Groups of Cubic and Quartic FieldsComplete determination of Galois groups in degrees three and four, where the discriminant and one cubic resolvent decide everything.2 min readRead MoreArticle7 Aug 2026Galois Groups of Quintic FieldsDetermining Galois groups in degree five, where solvability by radicals first fails and the resolvent degrees grow.2 min readRead MoreArticle7 Aug 2026Galois Groups of Sextic and Septic FieldsDegrees six and seven, where the number of candidate groups and the resolvent degrees make careful strategy essential.2 min readRead MoreArticle7 Aug 2026Test Polynomials for Galois Group SoftwareWhy Galois group implementations need a curated test set, what a good set covers, and how to build one.2 min readRead MoreArticle7 Aug 2026Constructing Tables of Number FieldsSystematically enumerating number fields of small degree and bounded discriminant, with canonical representatives and completeness arguments.2 min readRead MoreArticle7 Aug 2026Cyclic and Pure Cubic Field FamiliesCyclic cubic and pure cubic fields as parametrised families, with closed-form invariants that avoid general algorithms.2 min readRead MoreArticle7 Aug 2026Buchmann's Sub-exponential Algorithm: OverviewBuchmann's algorithm for class groups and units of arbitrary number fields, its structure, and where its cost concentrates.2 min readRead MoreArticle7 Aug 2026Ideal Reduction in Number FieldsFinding a small ideal in a given ideal class by lattice reduction, and why reduction is the enabling step for relation collection.2 min readRead MoreArticle7 Aug 2026Factor Base Selection and SmoothnessChoosing the factor base for class group computation, the smoothness trade-off, and how base size interacts with the linear algebra.2 min readRead MoreArticle7 Aug 2026Relation Matrix ConstructionGenerating relations among ideal classes, assembling the sparse matrix, and knowing when enough relations have been collected.2 min readRead MoreArticle7 Aug 2026Regulator and Fundamental Unit RecoveryExtracting fundamental units and the regulator from the kernel of the relation matrix, and confirming the unit system is fundamental.2 min readRead MoreArticle7 Aug 2026Verifying Class Group and Regulator ResultsConfirming class group and regulator results against the analytic class number formula, and what such confirmation does and does not establish.2 min readRead MoreArticle7 Aug 2026The Sub-exponential Algorithm in PracticePractical considerations in running class group computations: parameter tuning, parallelism, precision management and diagnostics.2 min readRead MoreArticle7 Aug 2026Elliptic Curves: Basic DefinitionsWhat an elliptic curve is, why the group structure exists, and the three distinct roles curves play in this collection.2 min readRead MoreArticle7 Aug 2026Weierstrass Equations and InvariantsGeneral and short Weierstrass forms, the discriminant and j-invariant, and the transformations relating equivalent models.2 min readRead MoreArticle7 Aug 2026The Group Law on an Elliptic CurveThe chord-and-tangent addition law, its explicit formulas, and the coordinate systems that avoid inversion.2 min readRead MoreArticle7 Aug 2026Elliptic Integrals and Elliptic FunctionsThe analytic origin of elliptic curves in elliptic integrals, and the doubly periodic functions that invert them.2 min readRead MoreArticle7 Aug 2026Lattices, Complex Tori and the Weierstrass p-FunctionThe identification of elliptic curves over the complex numbers with complex tori, and the Weierstrass function that realises it.2 min readRead MoreArticle7 Aug 2026Isogenies and Endomorphism RingsIsogenies as maps of curves respecting the group law, and the two possible endomorphism rings over a field of characteristic zero.2 min readRead MoreArticle7 Aug 2026Complex Multiplication and Class NumbersThe link between curves with complex multiplication and class groups of imaginary quadratic orders, and the Hilbert class polynomial.2 min readRead MoreArticle7 Aug 2026Modular Equations and the j-InvariantModular functions, modular equations relating j-invariants of isogenous curves, and their use in locating isogenies.2 min readRead MoreArticle7 Aug 2026Zeta Functions of Elliptic CurvesCounting points over finite fields, the Hasse bound, and how local counts assemble into a global zeta function.2 min readRead MoreArticle7 Aug 2026L-Functions and the Birch-Swinnerton-Dyer ConjectureThe L-function of an elliptic curve, the Birch-Swinnerton-Dyer conjecture, and what can and cannot be computed about rank.2 min readRead MoreArticle7 Aug 2026Computing with Elliptic Curves over CPractical computation with elliptic curves over the complex numbers: periods, the torus map, and heights.2 min readRead MoreArticle7 Aug 2026Curve Reduction and Tate's AlgorithmReduction of an elliptic curve modulo a prime, the classification of bad reduction types, and Tate's algorithm.2 min readRead MoreArticle7 Aug 2026Schoof's Point Counting AlgorithmSchoof's polynomial-time algorithm for counting points on a curve over a finite field, and the SEA improvements.2 min readRead MoreArticle7 Aug 2026Primality Versus Factoring: Framing the ProblemsWhy proving compositeness is easy, proving primality is harder, and factoring is harder still — and what this asymmetry means in practice.2 min readRead MoreArticle7 Aug 2026Fermat and Strong Pseudoprime TestsThe Fermat test, its failure on Carmichael numbers, and the strong pseudoprime test that repairs it.2 min readRead MoreArticle7 Aug 2026Lucas Sequences and Lucas PseudoprimesLucas sequences, the Lucas probable prime test, and why it complements the strong pseudoprime test rather than duplicating it.2 min readRead MoreArticle7 Aug 2026The Baillie-PSW Compositeness TestThe Baillie-PSW test combining a strong base-two test with a strong Lucas test, and its status as the practical standard.2 min readRead MoreArticle7 Aug 2026The Pocklington-Lehmer N-1 Primality TestProving primality from a partial factorisation of one less than the candidate, and the certificate this produces.2 min readRead MoreArticle7 Aug 2026N+1 Tests and the Lucas-Lehmer TestPrimality tests using the factorisation of one more than the candidate, and the Lucas-Lehmer test for Mersenne numbers.2 min readRead MoreArticle7 Aug 2026Trial Division and Lehman's MethodTrial division as the first factoring step, its cost, and Lehman's improvement on Fermat's method.2 min readRead MoreArticle7 Aug 2026The Pollard Rho Factoring MethodPollard's rho method: cycle detection in a pseudorandom sequence, the birthday bound, and Brent's improvement.2 min readRead MoreArticle7 Aug 2026The Pollard p-1 MethodPollard's p-1 method, its dependence on the smoothness of the group order, and why that dependence is its fatal limitation.2 min readRead MoreArticle7 Aug 2026Shanks's Class Group Factoring MethodShanks's method factoring an integer by finding an ambiguous form in the class group of the corresponding discriminant.2 min readRead MoreArticle7 Aug 2026Shanks's Square Forms Factorisation (SQUFOF)SQUFOF: factoring by finding a square form in the cycle of an indefinite quadratic form, and why it excels for small inputs.2 min readRead MoreArticle7 Aug 2026The Modern Primality Testing LandscapeThe methods available for proving primality, their complexities, and which to use for a given size of candidate.2 min readRead MoreArticle7 Aug 2026Gauss Sums and Jacobi SumsCharacters, Gauss sums and Jacobi sums, and the properties that make them useful for primality testing.2 min readRead MoreArticle7 Aug 2026Structure of the Jacobi Sum Primality TestThe overall design of the Jacobi sum primality test, its two phases, and where its complexity comes from.2 min readRead MoreArticle7 Aug 2026Checking the Condition C_pThe central congruence condition of the Jacobi sum test, what it asserts, and how it is verified in practice.2 min readRead MoreArticle7 Aug 2026Implementing the Jacobi Sum TestPractical implementation of the Jacobi sum test: precomputation, cyclotomic arithmetic, and the final divisor search.2 min readRead MoreArticle7 Aug 2026The Goldwasser-Kilian Primality TestPrimality proving by elliptic curves with known point counts, its recursive certificate, and the point-counting bottleneck.2 min readRead MoreArticle7 Aug 2026Atkin-Morain Elliptic Curve Primality ProvingECPP: using complex multiplication to construct curves of known order, avoiding point counting entirely.2 min readRead MoreArticle7 Aug 2026Primality Certificates and Independent VerificationWhat a primality certificate is, why verification is cheaper than production, and what a certificate does and does not guarantee.2 min readRead MoreArticle7 Aug 2026Smoothness and Sub-exponential ComplexitySmooth numbers, the Dickman function, and how balancing smoothness probability against factor base size produces sub-exponential running times.2 min readRead MoreArticle7 Aug 2026The Continued Fraction Factorisation MethodCFRAC: generating small quadratic residues from the continued fraction expansion, and the congruence-of-squares framework it established.2 min readRead MoreArticle7 Aug 2026The Schnorr-Lenstra Class Group Factoring MethodFactoring via class groups of quadratic orders, and its place as the conceptual bridge to the elliptic curve method.2 min readRead MoreArticle7 Aug 2026Elliptic Curves Modulo NWorking with elliptic curves modulo a composite, why the group law fails, and why that failure is exactly what is wanted.2 min readRead MoreArticle7 Aug 2026Elliptic Curve Arithmetic Modulo NImplementing curve arithmetic over a composite modulus: coordinate systems, inversion handling, and Montgomery form.2 min readRead MoreArticle7 Aug 2026The Elliptic Curve Method: Stage OneECM stage one: multiplying a point by a highly smooth scalar to reach the identity in one component.2 min readRead MoreArticle7 Aug 2026ECM Stage Two and Practical TuningECM stage two, the large prime search, and how the two bounds are tuned together.3 min readRead MoreArticle7 Aug 2026Quadratic Sieve Factor Base SelectionChoosing the factor base for the quadratic sieve, the quadratic residue criterion, and the multiplier.2 min readRead MoreArticle7 Aug 2026The Quadratic Sieve: Sieving StageThe sieving stage: identifying smooth polynomial values in bulk using logarithm accumulation rather than trial division.2 min readRead MoreArticle7 Aug 2026The Multiple Polynomial Quadratic SieveMPQS: using many polynomials with short intervals to keep values small, and the self-initialising variant.2 min readRead MoreArticle7 Aug 2026The Quadratic Sieve: Linear Algebra StageFinding dependencies in the relation matrix over the field with two elements, and why this stage is the practical bottleneck.2 min readRead MoreArticle7 Aug 2026Number Field Sieve: Polynomial Selection and StructureHow the number field sieve achieves its complexity, why polynomial selection matters so much, and the role of number field arithmetic.3 min readRead MoreArticle7 Aug 2026Number Theory Software PackagesThe computer algebra systems and libraries implementing these algorithms, what each is suited to, and why implementing from scratch is usually the wrong choice.2 min readRead MoreArticle7 Aug 2026Published Tables of Fields and CurvesStandard tables of number fields, elliptic curves, class numbers and factorisations, and how to use them responsibly.2 min readRead MoreArticle7 Aug 2026Choosing an Algorithm: Decision GuideA consolidated decision guide across the main computational tasks in this collection.1 min readRead MoreArticle7 Aug 2026Modern Factoring Methods ComparedComparing factoring methods by target size, expected factor size and available hardware, with a practical sequencing recommendation.2 min readRead MoreArticle7 Aug 2026Implementation Pitfalls and Testing StrategyThe recurring implementation errors in this subject and the testing discipline that catches them.2 min readRead MoreArticle7 Aug 2026Further Reading and Source NotesHow this collection is organised, how to read it, and notes on the source material and its treatment.2 min readRead More