Finite 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.

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 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 field has, by definition, a commutative multiplication operation. 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). According to Wedderburn's little theorem, any finite division ring must be commutative, and hence a finite field. This result shows that the finiteness restriction can have algebraic consequences.

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