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Computational Algebraic Number Theory: Field Overview
What computational algebraic number theory actually computes, why the problems are hard, and how the subject's algorithms fit together.
Tag
Computational Number Theory
Every page in the KEVOS library tagged Computational Number Theory. 123 pages.
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Abelian Categories
Additive and abelian categories, the axioms, exactness in a general abelian category, the Freyd-Mitchell embedding theorem, and the standard examples.
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Adjoint Functors
Adjoint pairs, unit and counit, the tensor-hom adjunction, preservation of limits and colimits, and the exactness consequences that make adjointness central to homological algebra.
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Algebraic Numbers and Number Fields
Algebraic numbers and integers, minimal polynomials, number fields as finite extensions of Q, real and complex embeddings, the signature, and the primitive element theorem.
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Algorithms for Elliptic Curves over ℚ
Computing minimal Weierstrass models, Tate's algorithm for reduction type and conductor, torsion subgroup determination, heights, and descent for rank computation.
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Applications of the Künneth Formulas
Applications of the Kunneth and universal coefficient theorems to products of spaces, group cohomology of direct products, and the ring structure on cohomology.
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Applications of the LLL Algorithm
Practical applications of lattice reduction: integer kernel and image computation, integer relation detection, recovering minimal polynomials from numerical approximations, and …
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Baby-Step Giant-Step and Class Group Structure
The 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 gr…
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Categories and Morphisms
Categories, objects and morphisms, monomorphisms and epimorphisms defined by cancellation, isomorphisms, and why the arrow-theoretic definitions differ from the element-based ones.
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Chain Complexes and Homology
Chain and cochain complexes, cycles and boundaries, homology as a functor, and the abelian category of complexes.
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Chain Homotopy and the Comparison Theorem
Chain homotopy between chain maps, homotopy equivalence, the comparison theorem for projective resolutions, and independence of derived functors from the chosen resolution.
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Change of Rings
Restriction and extension of scalars, induced and coinduced modules, the change-of-rings theorems, and the associated spectral sequences.
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Class Group and Unit Computation in General Number Fields
The general sub-exponential algorithm for class groups and units: factor base selection, ideal reduction, relation collection, the relation matrix and its kernel, and verificati…
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Class Groups, Units and the Regulator
The ideal class group, Dirichlet's unit theorem, the regulator, Minkowski's bound, and the analytic class number formula used to verify computed values.
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Class Numbers of Imaginary Quadratic Fields
Computing 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 numbe…
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Classical Factoring: Trial Division, Fermat and Lehman
Trial division and wheel factorisation, Fermat's difference of squares method, Lehman's improvement, and the role of these methods as a preprocessing stage.
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Classical Primality Proofs: Pocklington and Lehmer
The Pocklington-Lehmer N−1 test, partial factorisation requirements, the N+1 test with Lucas sequences, and combined methods.
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Cofree Modules and Essential Extensions
Cofree modules via Hom from the ring, the adjunction producing enough injectives, essential extensions, and the existence and uniqueness of injective hulls.
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Cohomology of Finite Cyclic Groups
The periodic free resolution for a finite cyclic group, the resulting periodic cohomology, norm and difference maps, and Tate cohomology.
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Cohomology of Products and Coproducts of Groups
Cohomology 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.
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Completions of Filtrations and lim1
Inverse limits and their failure of exactness, the derived functor lim^1, the Mittag-Leffler condition, the Milnor sequence, and completion of filtered objects.
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Complex Multiplication and Class Fields
Complex multiplication, isogenies, the relation between CM curves and imaginary quadratic orders, Hilbert and Weber class polynomials, and the CM method for curve construction.
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Compositeness Tests: Fermat and Miller–Rabin
The 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.
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Computation and Authoritative Sources
Software for computing resolutions, Ext, Tor and group cohomology, together with the authoritative databases and the KEVOS policy on reproducing computed data.
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Computing Ext Groups
Techniques 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.
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Computing Galois Groups of Number Fields
The resolvent method for determining Galois groups, the Frobenius cycle-type approach via Dedekind's theorem, transitive group classification by degree, and test polynomials.
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Computing the Maximal Order: the Round 2 Algorithm
The Pohst-Zassenhaus theorem, the Dedekind criterion, the radical and the ring of multipliers, and the Round 2 algorithm for computing the ring of integers.
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Continued Fraction Expansions
Simple continued fractions, convergent recurrences and best-approximation properties, Lagrange's periodicity theorem, and the expansion of a square root used for Pell's equation…
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Convergence of Spectral Sequences
Convergence conditions for spectral sequences, first-quadrant and bounded cases, conditional convergence, and the failure modes including non-vanishing lim^1.
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Decomposition of Prime Numbers in Number Fields
Ramification indices and residue degrees, the fundamental identity, Dedekind's theorem relating prime decomposition to polynomial factorisation modulo p, and computing valuations.
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Definition of Group Homology and Cohomology
Group 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 seque…
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Definition of Lie Algebra Cohomology
Lie algebra cohomology as Ext over the enveloping algebra, invariants and coinvariants, the explicit description in degrees 0 and 1, and derivations.
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Derivations and the Semidirect Product
Derivations and principal derivations, the semidirect product, complements and their conjugacy, and the interpretation of H^1 as classifying complements.
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Derived and Stable Categories
The derived category, localisation at quasi-isomorphisms, triangulated structure, derived functors in the modern sense, and the stable module category.
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Direct Sums, Products and Split Sequences
Direct sums and direct products of modules, their universal properties, the splitting lemma and equivalent characterisations of split short exact sequences.
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Discriminants and Integral Bases
The field discriminant, integral bases, the index of an equation order, and why computing the maximal order reduces to factoring the polynomial discriminant.
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Double Complexes and Total Complexes
Double complexes, the sign convention, the total complex by sum or product, and the two filtrations that give rise to spectral sequences.
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Duality and Opposite Categories
The opposite category, the duality principle, dual pairs of notions in homological algebra, and the limits of formal duality.
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Elliptic Curve L-functions and the Birch–Swinnerton-Dyer Conjecture
The 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.
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Elliptic Curve Primality Proving
Goldwasser-Kilian and Atkin-Morain elliptic curve primality proving: the group order downstep, the CM method for avoiding point counting, certificate structure and verification.
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Elliptic Curves: Definitions and the Group Law
Weierstrass forms, the discriminant and j-invariant, the group law with explicit formulas, torsion, and the structure of the group of points over finite fields.
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Exact Couples and Spectral Sequences
Spectral sequences as successive approximations, pages and differentials, exact couples and their derivation, and the standard sources of spectral sequences.
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Ext via Projective and Injective Resolutions
Computing Ext by resolving either variable, the double complex proof that the two agree, and the practical consequences for choosing a computation.
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Extensions of Modules and the Baer Sum
Extensions of modules, equivalence of extensions, the Baer sum defined by pullback and pushout, and the resulting abelian group structure on Ext.
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Factorisation of Polynomials Modulo a Prime
Squarefree decomposition, distinct-degree factorisation by gcd with x^(q^d) − x, equal-degree splitting by Cantor-Zassenhaus, and Berlekamp's linear-algebra approach.
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Filtered Differential Objects
Filtrations of chain complexes, the associated graded object, the spectral sequence of a filtered complex, and the interpretation of the pages.
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Finiteness Conditions on Groups
Cohomological dimension of groups, the Stallings-Swan theorem, finiteness conditions FP_n and F_n, duality groups, and the role of torsion.
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Free and Projective Modules
Free modules and bases, projective modules and the lifting property, the equivalence with direct summands of free modules, and projective resolutions.
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