# Finite Fields

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

### Polynomials over Finite Fields

Factor factor a polynomial over a finite field

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

### 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

### Equations over Finite Fields

Solve solve polynomial equations over a finite field

FindInstance find solution instances in a finite field