Engineering

Engineering Mathematics13 Aug 2026 · 18 min read

Theorem Dependency Map

A dependency map for noncommutative ring theory: the prerequisites and consumers of Wedderburn-Artin, Schur, the Density Theorem, Hopkins-Levitzki, Nakayama, Krull-Schmidt, Bass…

Triangular Rings: Ideals, Modules and Chain Conditions

The triangular ring built from two rings and a bimodule: its multiplication, the complete description of its left, right and two-sided ideals, its Jacobson radical, and the theo…

Engineering Mathematics13 Aug 2026 · 22 min read

Twisted Laurent Series Division Rings

Formal Laurent series with the twisted multiplication x a = sigma(a) x form a division ring. Computing its centre gives the first known centrally infinite division ring and, whe…

Unipotent Elements and the Lie–Kolchin–Suprunenko Theorem

Unipotent and lambda-potent linear transformations, the identification of unipotent elements with p-elements in characteristic p, and the Lie-Kolchin-Suprunenko triangularisatio…

Engineering Mathematics13 Aug 2026 · 19 min read

Uniqueness in the Wedderburn–Artin Decomposition

Lam (3.8) and (3.9): decompositions of a ring into indecomposable two-sided ideals are unique as ideals, the isotypic sums of minimal left ideals are ideals that annihilate one …

Engineering Mathematics13 Aug 2026 · 23 min read

Units and Idempotents in Integral Group Rings

Character-theoretic arithmetic applied to integral group rings: central torsion units are trivial up to a root of unity, the centre of a finite group is determined by its integr…

Engineering Mathematics13 Aug 2026 · 20 min read

Von Neumann Regular Rings

Von Neumann regular rings: the five equivalent conditions of Lam's theorem, left-right symmetry, the sandwich between semisimple and semiprimitive rings, the theorem that noethe…

Engineering Mathematics13 Aug 2026 · 19 min read

Wedderburn’s Factorisation Theorem

Wedderburn's factorisation theorem: the minimal polynomial of an algebraic conjugacy class of a division ring splits into linear factors with roots drawn from that class, togeth…

Engineering Mathematics13 Aug 2026 · 18 min read

Wedderburn’s Little Theorem: Finite Division Rings Are Fields

Wedderburn's Little Theorem: every finite division ring is a field. The class-equation proof with cyclotomic polynomials, the corollary that finite subrings of division rings ar…

Engineering Mathematics13 Aug 2026 · 20 min read

Well-Ordered Subsets of Ordered Groups

Well-ordered subsets of a totally ordered set are characterised by the descending chain condition and by the existence of nondecreasing subsequences. In an ordered group they ar…

When Are eR and fR Isomorphic?

Two idempotents of a ring generate isomorphic principal right modules exactly when one is a product ab and the other is ba. This page proves that criterion, derives the induced …

When Is a Group Algebra a Local Ring?

When a group algebra is a local ring: the augmentation ideal of a p-group in characteristic p is nilpotent, the resulting equivalences for kG, the extension to group rings over …

Abelian Groups: Definitions, Properties and Examples

Abelian groups: axioms, standard examples, and why the commutative case is sufficient for computational number theory.

Engineering Mathematics13 Aug 2026 · 22 min read

Algebraic and Geometric Multiplicities of Eigenvalues

Algebraic against geometric multiplicity: degree of the characteristic polynomial, the bound on eigenspace dimension, eigenvalue counts and defective matrices.

Algebraic Numbers and Minimal Polynomials

Algebraic numbers, minimal polynomials, algebraic integers, and the computational tests that distinguish them.

Algebras over a Ring

Algebras as rings carrying a compatible module structure, and the examples that matter computationally.

Algorithm Notation and Complexity Conventions

The notation, cost model and complexity conventions used throughout this collection, including the sub-exponential L-function.

Analysis of Berlekamp's Algorithm

Cost analysis of Berlekamp's algorithm and the variants that improve its dependence on field size.

Analysis of the Cantor-Zassenhaus Algorithm

The complete Cantor-Zassenhaus factorisation algorithm, its three stages, and its overall cost.

Applications of the Hermite Normal Form

Solving integer linear systems, computing indices and intersections, and testing module membership using the Hermite normal form.

Approximating Functions by Random Sampling

Estimating a quantity by sampling, the sample size required, and the resulting confidence guarantee.

Arbitrary Precision Arithmetic in Practice

Implementation concerns for multiprecision arithmetic: memory management, algorithm dispatch, and constant-time requirements.

Engineering Mathematics13 Aug 2026 · 26 min read

Archetypes: Reference Catalogue of Worked Systems

A catalogue of canonical worked linear systems, matrices and transformations, each fully analysed, designed so paired entries isolate a single structural variable.

Arithmetic Functions and Mobius Inversion

Multiplicative arithmetic functions, Dirichlet convolution, the Mobius function and the inversion formula.

Asymptotic Cost of Integer Multiplication

The M(n) abstraction, the hierarchy of multiplication algorithms, and why downstream bounds are quoted in terms of M(n) rather than fixed exponents.

Asymptotic Notation for Algorithm Analysis

Big-O, Omega, Theta and little-o notation, the conventions that make them precise, and the pitfalls of using them carelessly.

Atkin-Morain Elliptic Curve Primality Proving

ECPP: using complex multiplication to construct curves of known order, avoiding point counting entirely.

Engineering Mathematics13 Aug 2026 · 25 min read

Bases

A basis is a linearly independent spanning set: enough raw material to build every vector, and no more. Standard bases, unique coordinates and verification.

Engineering Mathematics13 Aug 2026 · 27 min read

Bases and Nonsingular Matrices

The columns of a square matrix form a basis of C^m exactly when the matrix is nonsingular, and every vector then has a unique representation in that basis.

Engineering Mathematics13 Aug 2026 · 25 min read

Bases from Row Spaces

Stack a spanning set as the rows of a matrix and row-reduce: the non-zero rows are a basis for the span. The method, its canonical output and an equality test.

Basic Polynomial Arithmetic

Representation of polynomials, addition, multiplication and division costs, and the dense versus sparse choice.

Basic Properties of Polynomial Rings

Degree, leading coefficients, the ring structure of R[X], and when it is an integral domain.

Berlekamp's Factorization Algorithm

Berlekamp's algorithm: the Berlekamp subalgebra, the kernel computation, and splitting by gcds.

Bertrand's Postulate

Bertrand's postulate that a prime always lies between n and 2n, its elementary proof, and its use in algorithm analysis.

Better Smoothness Density Estimates

Refined estimates for smooth number density and how they determine optimal sieve parameters.

Binary Powering and Exponentiation Chains

Square-and-multiply exponentiation, left-to-right and right-to-left variants, windowing, and why exponentiation cost drives primality testing.

Binary Quadratic Forms and the Ideal Correspondence

The dictionary between binary quadratic forms and ideals of a quadratic order, and why the form language is computationally preferable.

Brute-Force Discrete Logarithm Search

Exhaustive search for discrete logarithms, its cost, and its role as the baseline against which other methods are measured.

Buchmann's Sub-exponential Algorithm: Overview

Buchmann's algorithm for class groups and units of arbitrary number fields, its structure, and where its cost concentrates.

Cantor-Zassenhaus Equal Degree Splitting

Splitting a product of irreducibles of equal degree by random elements, the probability analysis, and the characteristic two variant.

Engineering Mathematics13 Aug 2026 · 22 min read

Casting Out Vectors: Reducing a Spanning Set

Keep the vectors indexed by pivot columns and discard the rest: the basis-of-a-span theorem turns any generating set into a linearly independent one with the same span.

Engineering Mathematics13 Aug 2026 · 24 min read

Characterization of Vector Spaces

Every vector space of dimension n is isomorphic to complex n-space, and two finite-dimensional spaces are isomorphic exactly when their dimensions agree.

Chebyshev's Theorem on the Density of Primes

Chebyshev's elementary bounds on the prime counting function, the binomial coefficient argument, and what they establish short of the prime number theorem.

Checking the Condition C_p

The central congruence condition of the Jacobi sum test, what it asserts, and how it is verified in practice.

Chinese Remainder Theorem Algorithms

Reconstructing an integer from residues, Garner's incremental method, and CRT as a strategy for controlling coefficient growth.

Chinese Remaindering and Polynomial Interpolation

The Chinese remainder theorem for polynomials, and Lagrange interpolation as its special case.

Choosing an Algorithm: Decision Guide

A consolidated decision guide across the main computational tasks in this collection.

Class Group and Unit Computation: the Computational Problem

Why class group and unit computation are a single problem, what the combined algorithm produces, and how the results are verified.