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 pⁿ elements; this field is denoted by or (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

[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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