Engineering Mathematics

The Quadratic Sieve: Linear Algebra Stage

Finding dependencies in the relation matrix over the field with two elements, and why this stage is the practical bottleneck.

The Quadratic Sieve: Sieving Stage

The sieving stage: identifying smooth polynomial values in bulk using logarithm accumulation rather than trial division.

The Regulator: Definition and Computation

The regulator as the covolume of the unit lattice, its computation, and the precision and verification it demands.

The Resolvent Method for Galois Groups

Constructing resolvent polynomials whose factorisation distinguishes candidate Galois groups, and the practical issues in using them.

The Round 2 Maximal Order Algorithm

The Round 2 algorithm: computing the maximal order prime by prime via radicals and rings of multipliers.

Engineering Mathematics13 Aug 2026 · 25 min read

The Row-Reducing Algorithm (Gauss-Jordan Elimination)

The algorithm that drives any matrix to reduced row-echelon form: pivot search, interchange, normalisation, column clearing, cost and pivoting strategy.

The RSA Cryptosystem

RSA key generation, encryption, decryption and correctness, together with the assumptions its security depends on.

The Schnorr-Lenstra Class Group Factoring Method

Factoring via class groups of quadratic orders, and its place as the conceptual bridge to the elliptic curve method.

The Sieve of Eratosthenes

The classical sieve for enumerating primes, its complexity, segmented variants, and its role as a precomputation step.

The Smith Normal Form Algorithm

The Smith normal form, its computation by alternating row and column reduction, and the invariant factors it exposes.

Engineering Mathematics13 Aug 2026 · 21 min read

The Span of a Set in a Vector Space

Linear combinations and spans in an abstract vector space: the definition, the proof that a span is always a subspace, and how membership becomes a linear system.

The Standard Representation of Algebraic Numbers

Representing field elements as coefficient vectors relative to a power basis or integral basis, with a common denominator.

The Structure of Finite Abelian Groups

The structure theorem decomposing every finite abelian group into cyclic factors, and its computational consequences.

The Structure of the Group of Units Modulo n

The structure of Z_n* as a product of cyclic groups, the Carmichael function, and why prime moduli behave differently.

The Sub-exponential Algorithm in Practice

Practical considerations in running class group computations: parameter tuning, parallelism, precision management and diagnostics.

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.