Finite fields, also known as Galois fields, are used in algebraic computation, error-correcting codes, cryptography, combinatorics, algebraic geometry, number theory and finite geometry. The Wolfram Language provides a complete suite of functions for working with finite fields, along with state-of-the-art algorithms for polynomial computation, equation solving and matrix operations in such fields.

Finite Fields and Field Embeddings

FiniteField — represent a finite field

FiniteFieldElement — represent an element of a finite field

FiniteFieldEmbedding — an embedding of a finite field in another finite field

FrobeniusAutomorphism — Frobenius automorphism of a finite field

ToFiniteField, FromFiniteField — convert expressions to and from finite field versions

FiniteFieldIndex, FromFiniteFieldIndex — convert to and from the index representation

FiniteFieldElementTrace  ▪  FiniteFieldElementNorm  ▪  MinimalPolynomial  ▪  MultiplicativeOrder  ▪  FiniteFieldElementPrimitiveQ

Polynomials over Finite Fields

Factor — factor a polynomial over a finite field

PolynomialGCD — find the GCD of polynomials with coefficients from a finite field

FactorList  ▪  FactorSquareFree  ▪  FactorSquareFreeList  ▪  PolynomialLCM  ▪  PolynomialExtendedGCD  ▪  Expand  ▪  Together  ▪  Cancel  ▪  IrreduciblePolynomialQ  ▪  PolynomialQuotient  ▪  PolynomialRemainder  ▪  PolynomialQuotientRemainder  ▪  Resultant  ▪  Discriminant

Linear Algebra over Finite Fields

Det — compute the determinant of a matrix with finite field element entries

Inverse — compute the inverse of a matrix with finite field element entries

LinearSolve — solve matrix equations over a finite field

Dot  ▪  RowReduce  ▪  NullSpace  ▪  MatrixRank  ▪  LUDecomposition  ▪  CharacteristicPolynomial

Equations over Finite Fields

Solve — solve polynomial equations over a finite field

FindInstance — find solution instances in a finite field

Reduce  ▪  SolveValues  ▪  Resolve