Engineering

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.

Engineering Mathematics13 Aug 2026 · 7 min read

Computing Modular Square Roots

Algorithms 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 lemm…

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.

Congruences and Quotient Algebras

Congruences as equivalence relations compatible with the operations, the construction of quotient algebras, and why congruences rather than subobjects are the general quotient d…

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…

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.

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.

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…

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.

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.

Derived and Stable Categories

The derived category, localisation at quasi-isomorphisms, triangulated structure, derived functors in the modern sense, and the stable module category.

Derived Functors

Left and right derived functors, their construction from resolutions, the fundamental properties, and the axiomatic characterisation by universality.

Engineering Mathematics13 Aug 2026 · 7 min read

Deterministic Primality Testing: the AKS Algorithm

The 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 improvement…

Direct Products, Factor Congruences and Direct Indecomposability

Direct products, projection homomorphisms, factor congruences as the internal signature of a decomposition, directly indecomposable algebras and the limits of unique factorisation.

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.

Directly Representable Varieties

Directly representable varieties, McKenzie's theorem that they are congruence-permutable, the classification of their directly indecomposable members, and the connection to the …

Discrete Probability for Algorithm Analysis

Finite probability distributions, conditional probability and independence, random variables, expectation and variance, Chebyshev and Chernoff bounds, the birthday paradox, hash…

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.

Discriminator Varieties

The ternary discriminator term, discriminator varieties, the Bulman-Fleming–Keimel–Werner representation theorem, and the exceptional package of structural properties that f…

Distributive and Modular Lattices: the M5 and N5 Criteria

Distributive 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.

Divisibility, Ideals and Unique Factorization

How divisibility, division with remainder, ideals, greatest common divisors and the fundamental theorem of arithmetic fit together — the structural foundation for every modular …

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.

Duality and Opposite Categories

The opposite category, the duality principle, dual pairs of notions in homological algebra, and the limits of formal duality.

Elementary Substructures and the Lowenheim-Skolem Theorems

Elementary substructures and extensions, the Tarski–Vaught test, the downward and upward Löwenheim–Skolem theorems, and the Skolem paradox.

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.

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.

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.

Equational Logic and the Completeness Theorem

The formal system of equational logic, its five inference rules, soundness, and Birkhoff's completeness theorem identifying derivability with semantic consequence.

Equivalence Relations and the Partition Lattice

The lattice of equivalence relations on a set, its identification with the partition lattice, why joins require alternating composites, and permutability as the condition that m…

Euclid's Algorithm and Modular Computation

The Euclidean and extended Euclidean algorithms with complexity analysis, computing modular inverses, Chinese remaindering in practice, multi-modular computation, and rational r…

Euler's Phi Function and Fermat's Little Theorem

Euler'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 theo…

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.

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.

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.

Factoring Polynomials over Finite Fields

Squarefree decomposition, distinct-degree and equal-degree factorization, the Cantor–Zassenhaus algorithm, Berlekamp's linear-algebra method, irreducibility testing and construc…

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.

Filtered Differential Objects

Filtrations of chain complexes, the associated graded object, the spectral sequence of a filtered complex, and the interpretation of the pages.

Engineering Mathematics13 Aug 2026 · 11 min read

Filters, Ideals and Ultrafilters

Filters and ideals on a Boolean algebra, principal and free filters, ultrafilters and their three equivalent characterisations, and the role of ultrafilters throughout the rest …

Engineering Mathematics13 Aug 2026 · 7 min read

Finite Fields: Existence, Uniqueness and Structure

Why 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.

Engineering Mathematics13 Aug 2026 · 11 min read

Finite State Acceptors and Recognisable Languages

Finite state acceptors as unary algebras, recognisable languages, the Myhill–Nerode congruence, the minimal automaton as a quotient, and the closure properties that follow.

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.

First-order Languages and Structures

First-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 i…

Engineering Mathematics13 Aug 2026 · 11 min read

Free Algebras and the Universal Mapping Property

Free algebras over a class, their construction as quotients of the term algebra, the universal mapping property, and the fact that F_K(X) lies in the variety generated by K.

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.

Fully Invariant Congruences and Equational Theories

Equational theories as fully invariant congruences on the term algebra, the dual isomorphism with the lattice of varieties, and equational bases.

Functionally Complete Algebras

Functionally complete algebras, the polynomial clone, the characterisation via simplicity and the discriminator as a polynomial, and the contrast with primality.

Functors and Their Exactness Properties

Covariant and contravariant functors, composition, additive functors, faithful and full functors, and the exactness hierarchy that governs derived functors.

Generators and Discrete Logarithms in ℤ*p

Finding 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 establ…