In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod ${\displaystyle p}$ when ${\displaystyle p}$ is a prime number.

The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number ${\displaystyle p}$ and every positive integer ${\displaystyle k}$ there are fields of order ${\displaystyle p^{k}}$. All finite fields of a given order are isomorphic.

Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.

A finite field is a field that is a finite set; this means that it has a finite number of elements on which multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the field axioms.

The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order ${\displaystyle q}$ exists if and only if ${\displaystyle q}$ is a prime power ${\displaystyle p^{k}}$ (where ${\displaystyle p}$ is a prime number and ${\displaystyle k}$ is a positive integer). In a field of order ${\displaystyle p^{k}}$, adding ${\displaystyle p}$ copies of any element always results in zero; that is, the characteristic of the field is ${\displaystyle p}$.

For ${\displaystyle q=p^{k}}$, all fields of order ${\displaystyle q}$ are isomorphic (see § Existence and uniqueness below). Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted ${\displaystyle \mathbb {F} _{q}}$, ${\displaystyle \mathbf {F} _{q}}$ or ${\displaystyle \mathrm {GF} (q)}$, where the letters GF stand for "Galois field".

In a finite field of order ${\displaystyle q}$, the polynomial ${\displaystyle X^{q}-X}$ has all ${\displaystyle q}$ elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.)

The simplest examples of finite fields are the fields of prime order: for each prime number ${\displaystyle p}$, the prime field of order ${\displaystyle p}$ may be constructed as the integers modulo ${\displaystyle p}$, ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$.