Finite field

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A finite field is a field with a finite number of elements; e,g, the fields \mathbb{F}_p := \mathbb{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 \mathbb{F}_{p^n} or GF_{p^n} (where GF stands for "Galois field").

Contents

Proofs of basic properties:

Finite characteristic:

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

Existence and uniqueness of Fp

To begin with it is follows by inspection that \mathbb{F}_p is a field. Furthermore, given any other field F' with p elements, one immidiately get an isomorphism F\to F' by mapping \sum_{i=1}^N 1_F \to \sum_{i=1}^N 1_{F'}.

Existence - general case

working over \mathbb{F}_p, let f(x) := x^{p^n}-x. Let F be the splitting field of f over \mathbb{F}_p. 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

Let F be a finite field of characteristic p, then it contains 0_F,1_F....\sum_{i=1}^{p-1}1_F; i.e. it contains a copy of \mathbb{F}_p. Hence, F is a vector field of finite dimension over \mathbb{F}_p. Moreover since the non 0 elements of F form a group, they are all roots of the polynomial x^{p^n-1}-1; hence the elements of F are all roots of f.

The Frobenius map

Let F be a field of characteritic p, then the map x\mapsto x^p is the generator of the Galois group Gal(F/\mathbb{F}_p).

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