Finite field

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

A finite field is a field with a finite number of elements; e,g, the fields 𝔽p:=ℤ/(p) (with the addition and multiplication induced from the same operations on the integers). For any primes number p, and natural number n, there exists a unique finite field with pn elements; this field is denoted by 𝔽pn or GFpn (where GF stands for "Galois field").

## Contents

* 1 Proofs of basic properties:
* 1.1 Finite characteristic:
* 1.2 Existence and uniqueness of Fp
* 1.3 Existence - general case
* 1.4 Uniqueness - general case
* 2 The Frobenius map

## Proofs of basic properties:[edit]

### Finite characteristic:[edit]

Let F be a finite field, then by the piegonhole principle there are two different natural numbers number n,m such that ∑i=1n1F=∑i=1m1F. hence there is some minimal natural number N such that ∑i=1N1F=0. Since F is a field, it has no 0 divisors, and hence N is prime.

### Existence and uniqueness of Fp[edit]

To begin with it is follows by inspection that 𝔽p is a field. Furthermore, given any other field F' with p elements, one immediately get an isomorphism F->F′ by mapping ∑i=1N1F->∑i=1N1F′.

### Existence - general case[edit]

working over 𝔽p, let f(x):=xpn−x. Let F be the splitting field of f over 𝔽p. 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.

### Uniqueness - general case[edit]

Let F be a finite field of characteristic p, then it contains 0F,1F....∑i=1p−11F; i.e. it contains a copy of 𝔽p. Hence, F is a vector field of finite dimension over 𝔽p. Moreover since the non 0 elements of F form a group, they are all roots of the polynomial xpn−1−1; hence the elements of F are all roots of f.

## The Frobenius map[edit]

Let F be a field of characteritic p, then the map x↦xp is the generator of the Galois group Gal(F/𝔽p).