Finite field

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

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 ${\displaystyle {\displaystyle \sum _{i=1}^n}{1}_F={\displaystyle \sum _{i=1}^m}{1}_F}$. hence there is some minimal natural number N such that ${\displaystyle {\displaystyle \sum _{i=1}^N}{1}_F=0}$. Since F is a field, it has no 0 divisors, and hence N is prime.

Existence and uniqueness of F_p[edit]

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

Existence - general case[edit]

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

Let F be a finite field of characteristic p, then it contains ${\displaystyle 0_F,1_F....{\displaystyle \sum _{i=1}^{p-1}}{1}_F}$; i.e. it contains a copy of ${\displaystyle {\mathbb{𝔽}}_p}$. Hence, F is a vector field of finite dimension over ${\displaystyle {\mathbb{𝔽}}_p}$. Moreover since the non 0 elements of F form a group, they are all roots of the polynomial ${\displaystyle x^{p^n-1}-1}$; 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 ${\displaystyle x\mapsto x^p}$ is the generator of the Galois group ${\displaystyle Gal(F/{\mathbb{𝔽}}_p)}$.