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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 ''n'' when ''n'' is a prime number. The number of elements of a finite field is called its ''order''. A finite field of order exists if and only if the order is a prime power (where is a prime number and is a positive integer). All fields of a given order are isomorphic. In a field of order , adding copies of any element always results in zero; that is, the characteristic of the field is . In a finite field of order , the polynomial 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 isn't required to have a commutative multiplication is called a division ring (or sometimes ''skewfield''). A finite division ring is a finite field by Wedderburn's little theorem. This result shows that the finiteness condition in the definition of a finite field can have algebraic consequences. 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. Finite fields appear in the following chain of class inclusions: == Definitions, first examples, and basic properties == A finite field is a finite set on which the four operations multiplication, addition, subtraction and division (excluding by zero) are defined, satisfying the rules of arithmetic known as the field axioms. The simplest examples of finite fields are the prime fields: for each prime number , the field (also denoted , , or ) of order (that is, size) is easily constructed as the . The elements of a prime field may be represented by integers in the range . The sum, the difference and the product are computed by taking the remainder by of the integer result. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see ). Let be a finite field. For any element in and any integer , let us denote by the sum of copies of . The least positive such that must exist and is prime; it is called the ''characteristic'' of the field. If the characteristic of is , the operation makes a -vector space. It follows that the number of elements of is . For every prime number and every positive integer , there are finite fields of order , and all these fields are isomorphic (see below). One may therefore identify all fields of order , which are therefore unambiguously denoted , or , where the letters GF stand for "Galois field".〔This notation was introduced by E. H. Moore in an address given in 1893 at the International Mathematical Congress held in Chicago .〕 The identity : is true (for every and ) in a field of characteristic . (This follows from the fact that all, except the first and the last, binomial coefficients of the expansion of are multiples of ). For every element in the prime field , one has (This is an immediate consequence of Fermat's little theorem, and this may be easily proved as follows: the equality is trivially true for and ; one obtains the result for the other elements of by applying the above identity to and , where successively takes the values modulo .) This implies the equality : for polynomials over . More generally, every element in satisfies the polynomial equation . 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Finite field」の詳細全文を読む スポンサード リンク
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