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 p^{k} (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 p^{k}, 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 X^{q} − 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 field GF(p) (also denoted Z/pZ, $\mathbb {F} _{p}$, or F_{p}) of order (that is, size) p may be constructed as the
p. 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 skewfield), however by
Wedderburn's little theorem, any finite division ring must be commutative, and hence a finite field.
The elements of such a field 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, let us denote by n ⋅ x the sum of n copies of x. The least positive n such that n ⋅ 1 = 0 must exist 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 $(k,x)\mapsto k\cdot x$ 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 p^{n} for some integer n.
For every prime number p and every positive integer n, there exist finite fields of order p^{n}, and all fields of this order are
isomorphic (see
§ Existence and uniqueness below). One may therefore identify all fields of order p^{n}, which are therefore unambiguously denoted $\mathbb {F} _{p^{n}}$, F_{pn} or GF(p^{n}), where the letters GF stand for "Galois field".^{
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The
identity
- $(x+y)^{p}=x^{p}+y^{p}$
(sometimes called the
freshman's dream) is true (for every x and y) in a field of characteristic p. (This follows from each
binomial coefficient of the expansion of (x + y)^{p}, except the first and the last, being a multiple of p).
For each element x in the field GF(p) for a prime number p, one has x^{p} = x (This is an immediate consequence of
Fermat's little theorem, and this may be proved as follows: the equality is trivially true for x = 0 and x = 1; one obtains the result for the other elements of GF(p) by applying freshman's dreams with y = 1 and x taking successively the values 1, 2, ..., p − 1 modulo p.) This implies the equality
- $X^{p}-X=\prod _{a\in {\rm {GF}}(p)}(X-a)$
for polynomials over GF(p). More generally, every element in GF(p^{n}) satisfies the polynomial equation x^{pn} − 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.