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Concept
Finite field
Summary
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.
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 p and every positive integer k there are fields of order all of which 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 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 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 = pk, all fields of order q are isomorphic (see 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 , 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.
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