Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains 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 given by the integers mod p when p is a prime number. In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains 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 given by the integers mod p when p is a prime number. 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 finite set which 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 the order q is a prime power pk (where p is a prime number and k is a positive integer). 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. If q = p k , {displaystyle q=p^{k},} 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}} , Fq or GF(q), where the letters GF stand for 'Galois field'. 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.) The simplest examples of finite fields are the fields of prime order: for each prime number p, the prime field of order p, denoted GF(p), Z/pZ, F p {displaystyle mathbb {F} _{p}} , or Fp, may be constructed as the integers modulo 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 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 {displaystyle (k,x)mapsto kcdot 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 pn for some integer n.