Engineering

Projective Classes of Epimorphisms

Projective classes, relative projectives, allowable epimorphisms, relative exactness, and the examples that motivate relative homological algebra.

Projective Modules over a Principal Ideal Domain

Structure of projective and finitely generated modules over a PID, the structure theorem, torsion and free parts, and the resulting short projective resolutions.

Pullbacks, Pushouts and General Limits

Pullbacks and pushouts, their construction in module categories, limits and colimits over a diagram, and exactness of filtered colimits.

Quadratic Fields and Binary Quadratic Forms

Quadratic fields, their discriminants and integral bases, prime decomposition by the Kronecker symbol, binary quadratic forms, reduction, and the correspondence between form cla…

Engineering Mathematics13 Aug 2026 · 7 min read

Quadratic Residues and Quadratic Reciprocity

Quadratic 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 re…

Quasigroups, Loops and Latin Squares

Quasigroups and loops as equationally defined algebras, the correspondence with Latin squares, why division must be in the type for quasigroups to form a variety, and the congru…

Quasiprimal Algebras and Pixley's Theorem

Quasiprimal algebras, Pixley's characterisation via inner isomorphisms, the relationship to primality, and the varieties quasiprimal algebras generate.

Real Quadratic Fields and the Infrastructure Method

Regulators and fundamental units of real quadratic fields, the continued fraction algorithm, the exponential cost problem, and Shanks's infrastructure with the distance function.

Recent Developments: the 1981 Frontier

The source's Recent Developments chapter: the commutator programme, the classification of varieties by congruence conditions, decidability questions, Boolean constructions, stru…

Reduced Products and Filtered Products

Reduced products over a filter, the congruence they induce, the direct product and ultraproduct as extreme cases, and the Horn sentence preservation theorem.

Relative Derived Functors

Construction of relative derived functors from relative resolutions, their properties, the comparison with absolute derived functors, and worked examples.

Representing Algebraic Numbers

The standard, matrix, conjugate-vector and minimal-polynomial representations of an algebraic number, their relative costs, and how to convert between them.

Resolutions for Group Cohomology

The bar resolution and its normalised version, cocycles and coboundaries in explicit terms, resolutions from presentations, and the trade-off between explicitness and size.

Engineering Mathematics13 Aug 2026 · 7 min read

Rings, Ideals and Quotient Rings

Commutative 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…

Root Finding over the Complex Numbers

Numerical root finding for polynomials over C: conditioning, the Newton and Aberth methods, splitting-circle approaches, and the precision management needed to support exact num…

Satellites

Left and right satellites of an additive functor, their construction, the connection with derived functors, and their use when the category lacks enough projectives.

Satisfaction, Truth and Elementary Equivalence

The satisfaction relation defined by recursion, truth for sentences, theories and models, elementary equivalence, and why elementarily equivalent structures can be non-isomorphic.

Semantic Embeddings and Undecidability

Semantic embeddings as a technique for transferring undecidability, the interpretation of graphs and semigroups into varieties, the decidability results that stand in contrast, …

Semisimple Lie Algebras and the Whitehead Lemmas

The 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.

Semisimple Varieties

Semisimple varieties, the relationship to discriminator varieties, residual smallness, and the structure available when subdirect decomposition terminates in simple algebras.

Shanks's SQUFOF Factoring Method

The 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.

Skew-free Algebras

Skew congruences in direct products, skew-free algebras and varieties, the relationship to congruence permutability and distributivity, and the role in Boolean representations.

Solving Polynomial Equations Modulo p

Root-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.

Square Roots Modulo a Prime

Computing 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.

Engineering Mathematics13 Aug 2026 · 6 min read

Steiner Triple Systems, Squags and Sloops

Steiner triple systems recast as algebras: squags (Steiner quasigroups) and sloops (Steiner loops), the equational axioms for each, and what the variety structure delivers that …

Engineering Mathematics13 Aug 2026 · 6 min read

Stone Duality and Boolean Spaces

Stone 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…

Subdirect Products and Birkhoff's Subdirect Representation Theorem

Subdirect products, subdirect irreducibility, the monolith, and Birkhoff's subdirect representation theorem with its Zorn's lemma proof.

Subexponential Factoring and Index Calculus

Smooth number densities, the index calculus method for discrete logarithms, the quadratic sieve and number field sieve for factoring, practical improvements, and what record com…

Subgroups: Restriction, Corestriction and Transfer

Restriction and corestriction (transfer) maps, the index formula, consequences for torsion, and detection on Sylow subgroups.

Subuniverses, Subalgebras and the Generation Operator

Subuniverses, subalgebras, the generation operator Sg(X), and the two equivalent descriptions of generation as intersection from above and as term-closure from below.

Syntactic Monoids and Kleene's Theorem

The syntactic congruence and syntactic monoid, Kleene's characterisation of recognisable languages as the regular ones, Schützenberger's star-free theorem, and Eilenberg's varie…

Tame Congruence Theory

Tame congruence theory: minimal sets, the five local types, type sets of varieties, and how the classification refines and largely supersedes the 1981 congruence-condition hiera…

Terms, Term Algebras and Term Operations

Terms 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 referen…

The Algebraic Approach to Constraint Satisfaction

The constraint satisfaction problem, polymorphisms as an algebraic invariant, the Feder–Vardi dichotomy conjecture, the algebraic reformulation, and the 2017 resolution.

The Boolean Prime Ideal Theorem

The Boolean Prime Ideal Theorem, its equivalent formulations, its strength relative to the axiom of choice, and which results in this collection depend on it.

The Center of an Algebra and Affine Representation

The centre of an arbitrary algebra, its first-order definition, the characterisation of algebras with Z(A) = ∇ as polynomially equivalent to modules, and the commutator programm…

The Chevalley–Eilenberg Resolution

The Chevalley-Eilenberg resolution of the trivial module, its differential, the resulting cochain complex, and the bound on cohomological dimension.

The Cohen–Lenstra Heuristics

The Cohen-Lenstra heuristics for class groups of quadratic fields, the weighting principle, predicted divisibility frequencies, and their role in validating computations.

The Compactness Theorem and its Consequences

The compactness theorem, its proof by ultraproducts, the standard consequences including non-standard models, and the systematic list of properties first-order logic cannot expr…

The Congruence Lattice Con A

The congruence lattice: completeness, algebraicity, its position inside Eq(A), simple and subdirectly irreducible algebras read off from it, and the classification of varieties …

The Continued Fraction Factoring Method

CFRAC: 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 t…

The Distribution of Primes

Chebyshev bounds, Bertrand's postulate, Mertens' theorems, the prime number theorem and primes in arithmetic progressions — the density results that make random prime generation…

The 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 …

Engineering Mathematics13 Aug 2026 · 11 min read

The Euclidean Algorithm and GCD Computation

Euclid's algorithm and its worst case, the binary GCD, Lehmer's method for multiprecision operands, and how to choose between them by operand size.

The Ext Functor

Ext defined via extensions, via projective resolutions and via injective resolutions, the agreement of the three definitions, and the basic vanishing criteria.

The Extended Euclidean Algorithm and Modular Inverses

The extended Euclidean algorithm, its loop invariants, half-extended variants, modular inversion, and rational reconstruction from a partial remainder sequence.

The Finite Basis Problem after Tarski

The 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 approa…

The Five-Term Exact Sequence

The five-term exact sequence for a normal subgroup, inflation and restriction maps, transgression, and its derivation from the Lyndon-Hochschild-Serre spectral sequence.