## Finite field |

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In **finite field** or **Galois field** (so-named in honor of *p**p* is a prime number.

Finite fields are fundamental in a number of areas of mathematics and computer science, including

- definitions, first examples, and basic properties
- existence and uniqueness
- explicit construction
- multiplicative structure
- frobenius automorphism and galois theory
- polynomial factorization
- applications
- extensions
- see also
- notes
- references
- external links

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 *p ^{k}* (where

In a finite field of order *q*, the *X ^{q}* −

A finite field is a finite set on which the four operations multiplication, addition, subtraction and division (excluding division by zero) are defined, satisfying the rules of arithmetic known as the *p*, the *p*) (also denoted **Z**/*p***Z**, , or **F**_{p}) of order (that is, size) *p* may be constructed as the

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 computed by taking the *p* of the integer result. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see

Let *F* be a finite field. For any element *x* in *F* and any *n*, denote by *n* ⋅ *x* the sum of *n* copies of *x*. The least positive *n* such that *n* ⋅ 1 = 0 exists and is a prime number; it is called the * characteristic* of the field.

If the characteristic of *F* is *p*, one can define multiplication of an element k of GF(*p*) by an element x of *F* by choosing an integer representative for k and using repeated addition. This multiplication makes *F* into a GF(*p*)-*F* is *p ^{n}* for some integer n.

For every prime number *p* and every positive integer *n*, there exist finite fields of order *p ^{n}*, and all fields of this order are

The

(sometimes called the *p* for every *x* and *y*. This follows from the *x* + *y*)^{p}, except the first and the last, is a multiple of *p*.

By *p*) then *x ^{p}* =

for polynomials over GF(*p*). More generally, every element in GF(*p*^{n}) satisfies the polynomial equation *x*^{pn} − *x* = 0.

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

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 *skew field*). By

Other Languages

العربية: حقل منته

беларуская: Канечнае поле

български: Крайно поле

català: Cos finit

čeština: Konečné těleso

Deutsch: Endlicher Körper

Ελληνικά: Πεπερασμένο σώμα

español: Cuerpo finito

français: Corps fini

한국어: 유한체

italiano: Campo finito

עברית: שדה סופי

Nederlands: Eindig lichaam (Ned) / Eindig veld (Be)

日本語: 有限体

norsk: Endelig kropp

polski: Ciało skończone

português: Corpo finito

română: Corp finit

русский: Конечное поле

Simple English: Galois field

српски / srpski: Коначно поље

suomi: Äärellinen kunta

svenska: Ändlig kropp

Türkçe: Sonlu alan

українська: Поле Галуа

اردو: متناہی میدان

粵語: 有限體

中文: 有限域