Galois fields

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.

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.)

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