# Finite field

A **finite field** is a field with a finite number of elements; e,g, the fields (with the addition and multiplication induced from the
same operations on the integers). For any prime number p, and natural number n, there exists a unique finite field with p^{n} elements; this
field is denoted by or (where GF stands for "Galois field").

## Contents |

## [edit] Proofs of basic properties:

### [edit] Finite characteristic:

Let *F* be a finite field, then by the pigeonhole principle there are two different natural numbers number *n, m* such that .
Hence there is some minimal natural number *N* such that . Since *F* is a field, it has no divisors of 0, and hence *N* is prime.

### [edit] Existence and uniqueness of F_{p}

To begin with it is follows by inspection that is a field. Furthermore, given any other field *F* with *p* elements, one immediately get an isomorphism by mapping for all *N*.

### [edit] Existence - general case

Working over , let . Let *F* be the splitting field of *f* over .
Note that *f*' = − 1, and hence the gcd of *f*, *f'* is 1, and all the roots of *f* in *F* are distinct. Furthermore, note that the set of roots of *f* is closed under addition and multiplication; hence
*F* is simply the set of roots of *f*.

### [edit] Uniqueness - general case

Let *F* be a finite field of characteristic *p*, then it contains ; i.e. it contains a copy of .
Hence, *F* is a vector field of finite dimension over . Moreover since the non-zero elements of *F* form a group, they are all roots of the polynomial
; hence the elements of *F* are all roots of this polynomial.

## [edit] The Frobenius map

Let *F* be a field of characteristic *p*, then the map is the generator of the Galois group .

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