Engineering Mathematics

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…

Gram–Schmidt Orthogonalisation

The Gram-Schmidt procedure, the mu coefficients and their role in reduction conditions, the numerical instability of the classical algorithm, and exact integral alternatives.

Group Extensions and H2

Group extensions with abelian kernel, factor sets and their equivalence, the classification by H^2, central extensions, and the obstruction interpretation.

H2, Hopf's Formula and the Schur Multiplier

Hopf'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.

Hensel Lifting and Factorisation over the Integers

Hensel's lemma and quadratic lifting, Mignotte's coefficient bounds, the exponential recombination problem, and the LLL-based polynomial-time solution.

Hilbert's Syzygy Theorem

Hilbert's chain-of-syzygies theorem, the Koszul resolution, global dimension of polynomial rings, and the historical and modern significance.

Homological Algebra and Algebraic Topology

The topological origins of homological algebra, singular and cellular chains, the Eilenberg-Zilber and Kunneth theorems in topology, classifying spaces, and the dictionary betwe…

Homological Algebra: Discipline Overview

Structural overview of homological algebra: the failure of exactness, resolutions and derived functors, the central invariants Ext and Tor, and how the machinery specialises to …

Homology of Small Categories

Homology of a small category with coefficients in a functor, the nerve and its geometric realisation, comparison with group homology, and the associated spectral sequences.

Homomorphisms and the Isomorphism Theorems

Homomorphisms, kernels, the first, second and third isomorphism theorems in arbitrary type, the correspondence theorem, and the congruence extension property.

Identities and Birkhoff's HSP Theorem

Identities, satisfaction, the operators M and Id as a Galois connection, and Birkhoff's HSP theorem identifying varieties with equationally definable classes, with proof.

Injective Modules and Dualization

Injective modules, the extension property, Baer's criterion, divisible groups, and the existence of enough injectives.

Injective Modules over a Principal Ideal Domain

Injective modules over a PID characterised by divisibility, the classification of injective abelian groups, and the resulting short injective resolutions.

Integer Square Roots and Perfect Power Detection

Computing exact integer square roots by Newton's method, fast rejection of non-squares by modular filters, and detection of perfect powers and prime powers.

Kan Extensions

Left and right Kan extensions, their construction by colimits and limits, the adjunction characterisation, and their role in homology of small categories.

Lattice Homomorphisms, Isomorphisms and Sublattices

Lattice homomorphisms, isomorphisms and sublattices: why every lattice homomorphism is monotone but not conversely, how to recognise a genuine sublattice, and the role of embedd…

Lattices and Quadratic Forms

Lattices as discrete subgroups, bases and unimodular change of basis, the Gram matrix and determinant, successive minima, and the correspondence with positive definite quadratic…

Legendre, Jacobi and Kronecker Symbols

Quadratic residues, Euler's criterion, the Legendre symbol, its Jacobi and Kronecker extensions, and the reciprocity-based algorithm that evaluates them in logarithmic time.

Lie Algebra Extensions and H2

Extensions of Lie algebras with abelian kernel, the classification by H^2, central extensions, and examples including the Heisenberg and affine algebras.

Lie Algebras and the Universal Enveloping Algebra

Lie algebras, representations as modules, the universal enveloping algebra and its universal property, the Poincare-Birkhoff-Witt theorem, and the augmentation ideal.

Linear Algebra Algorithms over Fields and Rings

Exact Gaussian elimination, fraction-free elimination, determinant and characteristic polynomial algorithms, and kernel and image computation over fields and over ℤ.