Engineering

The Sub-Resultant GCD Algorithm

The sub-resultant remainder sequence: predicting the divisible factor at each step to keep coefficients near minimal without content computation.

The Subfield Problem

Finding the subfields of a number field, by lattice methods and by linear algebra over the complex numbers.

Trace, Norm and the Characteristic Polynomial

Trace, norm and characteristic polynomial of a field element, their computation, and their use as invariants and cross-checks.

Trial Division and Basic Primality Testing

Trial division as a primality test and as a filter, its exponential cost, and the role it still plays in practice.

Trial Division and Lehman's Method

Trial division as the first factoring step, its cost, and Lehman's improvement on Fermat's method.

Trial Division up to a Small Bound

Choosing the trial division bound ahead of a probabilistic test, and the cost balance that determines it.

Unique Factorisation Domains, Content and Primitive Parts

Content and primitive part, Gauss's lemma, and why factoring over the rationals reduces to factoring over the integers.

Unique Factorization Domains

Unique factorisation domains, the distinction between irreducible and prime, and the standard examples and counterexamples.

Unique Factorization in Euclidean and Principal Ideal Domains

Why Euclidean domains are principal ideal domains and why principal ideal domains have unique factorisation.

Unique Factorization of Polynomials

Unique factorisation in polynomial rings over a field, and the extension to polynomial rings over a UFD.

Unique Factorization of the Integers

The fundamental theorem of arithmetic: existence and uniqueness of prime factorisation, and why the uniqueness half is the difficult one.

Uniqueness of Reduced Row-Echelon Form

Proof that a matrix has exactly one reduced row-echelon form: pivot columns agree by induction, ranks agree, and rows are forced to coincide entry by entry.

Useful Facts and Standard Estimates

The analytic inequalities, series estimates and elementary bounds relied on repeatedly in the analysis of number-theoretic algorithms.

Valuations and Uniformisers

Valuations at prime ideals, uniformising elements, and computing the exponent of a prime in an ideal factorisation.

Engineering Mathematics13 Aug 2026 · 21 min read

Vector Form of Solution Sets

Express every solution of a linear system as a fixed vector plus a linear combination of n-r vectors read directly from the reduced row-echelon form.

Engineering Mathematics13 Aug 2026 · 24 min read

Vector Representation

Vector representation relative to an ordered basis: the coordinate map is a well-defined, injective and surjective linear transformation onto complex n-space.

Engineering Mathematics13 Aug 2026 · 26 min read

Vector Space Axioms

The ten defining properties of a vector space: closure, commutativity, associativity, zero vector, additive inverses, distributivity and the unit scalar.

Engineering Mathematics13 Aug 2026 · 24 min read

Vector Space Properties of Column Vectors

The ten algebraic properties of column vector addition and scalar multiplication, how each is proved entrywise, and why they license later manipulation.

Vector Space Properties of Matrices

The ten vector space properties of matrix addition and scalar multiplication: closure, commutativity, associativity, the zero matrix, additive inverses and distributivity.

Vector Spaces and Dimension

Vector spaces over a field, the well-definedness of dimension, and the rank-nullity relation.

Verifying Class Group and Regulator Results

Confirming class group and regulator results against the analytic class number formula, and what such confirmation does and does not establish.

Weierstrass Equations and Invariants

General and short Weierstrass forms, the discriminant and j-invariant, and the transformations relating equivalent models.

What Is Linear Algebra?

What makes an equation linear, why flatness matters, and how addition and scalar multiplication alone generate the whole of linear algebra in any dimension.

Worked Examples: Computing Eigenvalues and Eigenvectors

A gallery of computed spectra: distinct, repeated, defective, complex and zero eigenvalues, with algebraic and geometric multiplicities for each eigenspace.

Z-Modules and Integer Matrix Problems

Finitely generated abelian groups as integer matrix problems, and the two normal forms that answer the two basic questions about them.

Zero Divisors and Integral Domains

Zero divisors, integral domains, and why the absence of zero divisors is what makes cancellation and root counting work.

Zeta Functions of Elliptic Curves

Counting points over finite fields, the Hasse bound, and how local counts assemble into a global zeta function.

Abelian Categories

Additive and abelian categories, the axioms, exactness in a general abelian category, the Freyd-Mitchell embedding theorem, and the standard examples.

Abelian Groups and Cyclic Structure

Abelian groups, subgroups, cosets and quotient groups, Lagrange's theorem, homomorphisms and isomorphism theorems, cyclic groups and the structure theorem for finite abelian gro…

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.

Engineering Mathematics13 Aug 2026 · 10 min read

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.

Algebras, Types and Signatures

Algebras of arbitrary type: signatures, arities, the significance of nullary operations, and how the choice of type determines subalgebras, homomorphisms and the whole subsequen…

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.

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.

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 …

Asymptotic Notation and Machine Models

Big-O, Omega and Theta notation, the RAM model, bit complexity versus operation counts, input size measured in bits, polynomial versus subexponential versus exponential time, an…

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…

Engineering Mathematics13 Aug 2026 · 6 min read

Boolean Algebras: Axioms and Structure

Boolean algebras as an equational class, the two-element algebra as the unique subdirectly irreducible member, atoms and atomlessness, and why the variety is arithmetical.

Engineering Mathematics13 Aug 2026 · 6 min read

Boolean Powers

Boolean powers A[B]*, their construction as locally constant functions on a Stone space, the identities they preserve, and filtered Boolean powers as the refinement that carries…

Engineering Mathematics13 Aug 2026 · 10 min read

Boolean Products

Boolean products as subdirect products over a Boolean space with a patching condition, the relationship to sheaves, the equaliser condition, and the classes of algebras admittin…

Engineering Mathematics13 Aug 2026 · 6 min read

Boolean Rings and the Boolean Algebra-Ring Correspondence

Boolean rings, the mutual translation with Boolean algebras, term equivalence as the precise relationship, and why the ring picture makes ideals and the prime spectrum available.

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.

Chain Complexes and Homology

Chain and cochain complexes, cycles and boundaries, homology as a functor, and the abelian category of complexes.

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.

Change of Rings

Restriction and extension of scalars, induced and coinduced modules, the change-of-rings theorems, and the associated spectral sequences.

Engineering Mathematics13 Aug 2026 · 10 min read

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…

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.

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…