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Properties

A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms.

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

If q = p, all fields of order 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 F q {\displaystyle \mathbb {F} _{q}}, F or GF(q), where the letters GF stand for "Galois field".

In a finite field of order q, the polynomial X − 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.)

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

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 the remainder of the division by p of the result of the corresponding integer operation. 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 is the characteristic p of the field. This allows defining a multiplication (k, x) ↦ k ⋅ x of an element k of GF(p) by an element x of F by choosing an integer representative for k. This multiplication makes F into a GF(p) - vector space. It follows that the number of elements of F is p for some integer n.

The identity (x + y) p = x p + y p {\displaystyle (x+y)^{p}=x^{p}+y^{p}} (sometimes called the freshman's dream) is true in a field of characteristic p. This follows from the binomial theorem, as each binomial coefficient of the expansion of (x + y), 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 x = x. This implies the equality X p − X = ∏ a ∈ G F (p) (X − a) {\displaystyle X^{p}-X=\prod _{a\in \mathrm {GF} (p)}(X-a)} for polynomials over GF(p). More generally, every element in GF(p) satisfies the polynomial equation x − 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 piece of 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.