Engineering Mathematics

Computing Modular Square Roots: Composite Modulus

Square roots modulo a composite, the four roots for a semiprime, and the equivalence with factoring.

Computing Modular Square Roots: Prime Modulus

Extracting square roots modulo a prime, the easy case for p congruent to 3 mod 4, and the randomised algorithm in general.

Computing Modular Square Roots: Prime Power Modulus

Lifting a square root from a prime to a prime power by Hensel's method, and the special handling powers of two require.

Computing Rank, Kernel and Image

Extracting rank, kernel basis and image basis from an echelon form, and the applications to polynomial factorisation.

Engineering Mathematics7 Aug 2026 · 21 min read

Computing the Inverse of a Matrix

How to compute a matrix inverse: the two-by-two determinant formula, the column-by-column view, and Gauss-Jordan reduction of the augmented array [A | I].

Computing the Jacobi Symbol

The Euclid-style algorithm for evaluating a Jacobi symbol in quadratic time without factoring either argument.

Computing the Structure of Residue Rings

The structure of the unit group of a residue ring of the maximal order, and its computation by decomposition and lifting.

Computing with Elliptic Curves over C

Practical computation with elliptic curves over the complex numbers: periods, the torus map, and heights.

Conditional Probability and Independence

Conditional probability, Bayes' theorem, independence and the distinction between pairwise and mutual independence.

Congruences and Modular Arithmetic

Congruence as an equivalence relation compatible with arithmetic, and the basic manipulation rules including where cancellation fails.

Engineering Mathematics7 Aug 2026 · 23 min read

Conjugates of Complex Numbers

The complex conjugate: definition, the theorems that it respects addition and multiplication, involution, conjugate root pairs and its role in adjoints and inner products.

Conjugates, Norms and Traces

Conjugates of a finite field element, the norm and trace maps, and their surjectivity onto the subfield.

Consequences of Unique Factorization

What follows from the fundamental theorem: gcd and lcm via exponents, irrationality proofs, divisor counting and multiplicative structure.

Consistent and Inconsistent Systems

How to decide whether a linear system has a solution: the augmented-column pivot test, the roles of r, D and F, and why a system has zero, one or infinitely many solutions.

Constructing Tables of Number Fields

Systematically enumerating number fields of small degree and bounded discriminant, with canonical representatives and completeness arguments.

Continued Fraction Expansion of Real Numbers

Continued fractions, convergents as best rational approximations, and the periodic expansion of quadratic irrationals.

Cosets and Lagrange's Theorem

Cosets as a partition of a group, Lagrange's theorem, and its consequences for element orders.

Curve Reduction and Tate's Algorithm

Reduction of an elliptic curve modulo a prime, the classification of bad reduction types, and Tate's algorithm.

Cyclic and Pure Cubic Field Families

Cyclic cubic and pure cubic fields as parametrised families, with closed-form invariants that avoid general algorithms.

Cyclic Groups

Cyclic groups, their generators, subgroup structure, and the criterion for the units modulo n to be cyclic.

Detecting Algebraic and Linear Dependence with LLL

Recovering exact integer relations from numerical approximations using LLL, and the precision requirements that make the method reliable.

Determinant Computation Strategies

Computing exact determinants by fraction-free elimination, modular methods with Hadamard bounds, and how to choose between them.

Deterministic Polynomial Factorization Algorithms

Deterministic factorisation over finite fields, the obstacles, and what is known conditionally.

Deterministic Primality Testing: The Basic Idea

The polynomial identity underlying AKS primality testing and the obstacle that makes it non-trivial to exploit.

Engineering Mathematics7 Aug 2026 · 24 min read

Diagonalization

When a square matrix is similar to a diagonal matrix: the eigenvector basis test, the multiplicity criterion, distinct eigenvalues, and fast matrix powers.

Engineering Mathematics7 Aug 2026 · 25 min read

Dimension

Dimension of a vector space defined as the size of any basis, with the spanning-sets theorem and the proof that all bases of a space have identical size.

Engineering Mathematics7 Aug 2026 · 25 min read

Dimension of Common Vector Spaces

Dimensions of the standard vector spaces: column vectors, polynomials of bounded degree and matrices, plus subspace dimension and infinite-dimensional examples.

Discrete Logarithms in Groups of Prime Power Order

Solving discrete logarithms in a group of prime power order by digit-by-digit lifting.

Discrete Logarithms in the Full Group Modulo p

The Pohlig-Hellman reduction combining prime power subproblems by the Chinese remainder theorem.

Discriminants and Integral Bases

The discriminant of a basis, the field discriminant, and the index-squared relation that governs maximal order computation.

Distinct Degree Factorisation

Separating irreducible factors by degree using GCDs against Frobenius powers, and the early-abort strategies that make it fast.

Distinct Degree Factorization

Separating the irreducible factors of a polynomial by degree using gcds with Frobenius powers.

Engineering Mathematics7 Aug 2026 · 4 min read

Divisibility and Primality

Divisibility of integers, the definition of primes and composites, and the basic properties that support every later result in elementary number theory.

Division with Remainder for Integers

The division algorithm for integers, the uniqueness of quotient and remainder, and the role of well-ordering in establishing it.

ECM Stage Two and Practical Tuning

ECM stage two, the large prime search, and how the two bounds are tuned together.

Engineering Mathematics7 Aug 2026 · 23 min read

Eigenvalues and Eigenvectors of a Matrix

Definition of eigenvalues and eigenvectors, why the eigenvector must be non-zero while the eigenvalue may be zero, and how eigenvectors for one eigenvalue form a subspace.

Eigenvalues and Eigenvectors of Linear Transformations

Eigenvalues and eigenvectors of a linear transformation of a vector space to itself: eigenspaces, the meaning of zero, and the link to matrix representations.

Eigenvalues of Hermitian Matrices

Hermitian and real symmetric matrices have real eigenvalues and orthogonal eigenvectors: inner-product proofs, orthonormal modal bases and engineering use.

Engineering Mathematics7 Aug 2026 · 24 min read

Elementary Row Operations and Row Equivalence

The three elementary row operations, their shorthand notation, why each is reversible, and how row equivalence groups matrices with identical solution sets.

Elliptic Curve Arithmetic Modulo N

Implementing curve arithmetic over a composite modulus: coordinate systems, inversion handling, and Montgomery form.

Elliptic Curves Modulo N

Working with elliptic curves modulo a composite, why the group law fails, and why that failure is exactly what is wanted.

Elliptic Curves: Basic Definitions

What an elliptic curve is, why the group structure exists, and the three distinct roles curves play in this collection.

Elliptic Integrals and Elliptic Functions

The analytic origin of elliptic curves in elliptic integrals, and the doubly periodic functions that invert them.

Engineering Mathematics7 Aug 2026 · 4 min read

Equal Degree Factorization

Splitting a product of same-degree irreducible factors by random splitting, and the success probability.

Engineering Mathematics7 Aug 2026 · 33 min read

Equivalences of Nonsingular Matrices

Thirteen equivalent conditions on a square matrix: nonsingular, row-reduces to the identity, trivial null space, invertible, full rank, nonzero determinant.

Equivalent Systems and Equation Operations

Two systems are equivalent when their solution sets agree. The three equation operations, the theorem that they preserve solutions, and why the scalar must be non-zero.

Error-Correcting Codes and Algebraic Decoding

Reed-Solomon codes, their distance property, and decoding by rational function reconstruction.

Essential Discriminant Divisors

Primes that divide the index for every choice of defining polynomial, why they exist, and what they force computationally.