Guides

Finite Fields: Preliminaries

The characteristic of a finite field, why its order is a prime power, and the prime subfield.

Finite Probability Distributions

Finite sample spaces, probability distributions, events and the basic laws governing them.

Flipping a Coin Until a Head Appears

The geometric waiting time, its expectation and tail, and its role as the model for repeat-until-success algorithms.

Formal Derivatives of Polynomials

The formal derivative as an algebraic operation, its rules, and its use in detecting repeated factors.

Formal Laurent Series

Formal Laurent series, the field of fractions of the power series ring, and the valuation structure.

Formal Power Series

Formal power series, their arithmetic, invertibility criterion, and use as generating functions.

Galois Groups of Cubic and Quartic Fields

Complete determination of Galois groups in degrees three and four, where the discriminant and one cubic resolvent decide everything.

Galois Groups of Quintic Fields

Determining Galois groups in degree five, where solvability by radicals first fails and the resolvent degrees grow.

Galois Groups of Sextic and Septic Fields

Degrees six and seven, where the number of candidate groups and the resolvent degrees make careful strategy essential.

Gauss Sums and Jacobi Sums

Characters, Gauss sums and Jacobi sums, and the properties that make them useful for primality testing.

Gaussian Elimination

Gaussian elimination over a field, its complexity, pivoting, and its role as the bottleneck in sieve algorithms.

Gaussian Elimination and Linear Systems

Exact Gaussian elimination, pivoting for entry growth rather than stability, and the fraction-free Bareiss variant.

Gaussian Elimination over Finite Fields

Elimination over a finite field, structured methods for very large sparse systems, and why the linear algebra stage limits sieving methods.

General Properties of Extension Fields

Field extensions, degree, algebraic elements and minimal polynomials.

Generating a Random Factored Number

Producing a uniformly random integer together with its complete factorisation, in polynomial time, without factoring.

Generating a Random k-Bit Prime

Generating a prime of exactly k bits, the density in that range, and the constraints imposed by cryptographic use.

Generating a Random k-Bit Prime with Miller-Rabin

Assembling bit-length constraint, trial division filtering and Miller-Rabin into a complete prime generator.

Generating a Random Non-Increasing Sequence

Sampling a random non-increasing sequence in a bounded range, and its role as a subroutine in generating factored numbers.

Generating a Random Number from a Given Interval

Sampling uniformly from an arbitrary range using a source of random bits, and controlling the resulting bias.

Generating a Random Prime

Generating a random prime by repeated candidate testing, the expected number of trials, and the sieving optimisation.

Generating a Random Prime Between 2 and M

Generating a uniform random prime below a bound, the analysis of the retry loop, and the resulting output distribution.

Group Homomorphisms and Isomorphisms

Structure-preserving maps between groups, isomorphisms, and what it means for two groups to be the same.

Hash Function Families

Families of hash functions, keyed selection, and the properties required of them in algorithm design.

Hash Tables

Hash tables analysed with universal families: expected chain length, load factor and collision resolution.

Hensel Lifting for Polynomial Factors

Lifting a factorisation modulo a prime to a factorisation modulo a prime power, the quadratic variant, and the coprimality condition.

Engineering Mathematics13 Aug 2026 · 21 min read

Homogeneous Systems of Equations

Linear systems with every constant term zero: why they are always consistent, what the trivial solution is, and why more unknowns than equations guarantees infinitely many solut…

Ideal Multiplication and Division

Multiplying, inverting and dividing ideals as module operations, and controlling the growth these operations cause.

Ideal Norm Computation

The norm of an ideal as its index in the order, its multiplicativity, and its use as a size measure and consistency check.

Ideal Reduction in Number Fields

Finding a small ideal in a given ideal class by lattice reduction, and why reduction is the enabling step for relation collection.

Ideal Representation by Two Elements

Representing an ideal by two generators, why two always suffice, and the trade-off against the canonical matrix form.

Engineering Mathematics13 Aug 2026 · 9 min read

Ideals and Greatest Common Divisors of Integers

Greatest common divisors defined through ideals, Bezout's identity, and why the ideal-theoretic view is the one that generalises.

Ideals and Quotient Rings

Ideals, principal ideal domains, quotient rings, and the condition under which a quotient is a field.

Ideals of the Maximal Order

Ideals and fractional ideals, unique factorisation into primes, and the group structure that makes the class group possible.

Imaginary Quadratic Class Numbers by Counting Reduced Forms

Computing class numbers and group structure for imaginary quadratic fields by enumerating reduced forms, and where the method runs out.

Implementation Pitfalls and Testing Strategy

The recurring implementation errors in this subject and the testing discipline that catches them.

Implementing the Jacobi Sum Test

Practical implementation of the Jacobi sum test: precomputation, cyclotomic arithmetic, and the final divisor search.

Infinite Discrete Probability Distributions

Countably infinite sample spaces, convergence conditions, and the geometric distribution arising from unbounded loops.

Engineering Mathematics13 Aug 2026 · 21 min read

Injective Linear Transformations

Injective (one-to-one) linear transformations: the defining condition, how to prove injectivity by reducing to a homogeneous system, and how to disprove it.

Engineering Mathematics13 Aug 2026 · 22 min read

Injective Transformations, Independence and Dimension

How injective linear transformations preserve linear independence, why the images of a basis decide injectivity, and why domain dimension cannot exceed codomain.

Engineering Mathematics13 Aug 2026 · 20 min read

Inner Products of Vectors

The complex inner product of two column vectors: its definition with a conjugate on the second argument, additivity, scalar rules, conjugate symmetry and positivity.