Definitions, first examples, and basic properties
The number of elements of a finite field is called its order. A finite field of order q exists if and only if the order q is a prime power pk (where p is a prime number and k is a positive integer). All finite fields of a given order are isomorphic. In a field of order pk, adding p copies of any element always results in zero; that is, the characteristic of the field is p.
In a finite field of order q, the polynomial Xq − X has all 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.)
A finite field is a finite set on which the four operations multiplication, addition, subtraction and division (excluding division by zero) are defined, satisfying the rules of arithmetic known as the field axioms. The simplest examples of finite fields are the fields of prime order: for each prime number p, the prime field GF(p) (also denoted Z/pZ, , or Fp) of order (that is, size) p may be constructed as the p.
The elements of the prime field of order p may be represented by integers in the range 0, ..., p − 1. The sum, the difference and the product are computed by taking the remainder by p of the integer result. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm § Modular integers).
Let F be a finite field. For any element x in F and any integer n, denote by n ⋅ x the sum of n copies of x. The least positive n such that n ⋅ 1 = 0 exists and is a prime number; it is called the characteristic of the field.
If the characteristic of F is p, one can define multiplication of an element k of GF(p) by an element x of F by choosing an integer representative for k and using repeated addition. This multiplication makes F into a GF(p)-vector space. It follows that the number of elements of F is pn for some integer n.
For every prime number p and every positive integer n, there exist finite fields of order pn, and all fields of this order are isomorphic (see § Existence and uniqueness below). One may therefore identify all fields of order pn, which are therefore unambiguously denoted , Fpn or GF(pn), where the letters GF stand for "Galois field".
(sometimes called the freshman's dream) is true in a field of characteristic p for every x and y. This follows from the binomial theorem, as each binomial coefficient of the expansion of (x + y)p, except the first and the last, is a multiple of p.
By Fermat's little theorem, if p is a prime number and x is in the field GF(p) then xp = x. This implies the equality
for polynomials over GF(p). More generally, every element in GF(pn) satisfies the polynomial equation xpn − x = 0.
Any finite field extension of a finite field is separable and simple. That is, if E is a finite field and F is a subfield of E, then E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon, finite fields are perfect.
A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes skew field). By Wedderburn's little theorem, any finite division ring is commutative, and hence is a finite field.