All finite fields are isomorphic to GF(p^n) (idea)

Every finite field F is isomorphic to GF(pⁿ) for some prime p and natural n. That is to say, F has pⁿ elements, and all fields with the same number of elements are isomorphic. (We then call the field with pⁿ elements GF(pⁿ). This is short for Galois Field.)

Proof: If a field F contains a field K, then F is a vector space over K. If we show that for every F there exists a prime p such that the set of elements of the form 1+1+...+1 is isomorphic to the field GF(p) -- the field defined by taking the addition and multiplication operations modulo p on the integers 0,1,...,p-1 -- then F is a finite vector space over GF(p). It therefore has a dimension n, and thus pⁿ distinct elements. Obviously, two fields F, F' with the same number of elements are isomorphic as vector spaces over GF(p) (all finite dimensional vector spaces over F re isomorphic to Fⁿ) . Such an isomorphism preserves addition (in the fields) as required, but not the fields' multiplication -- only multiplication by one of the p elements 1,1+1,... Choose an isomorphism of this sort which also preserves multiplication (just start by mapping the unit (1) of F to the unit of F' and proceed by extending the map any "legal" way...) and you have an isomorphism between the fields.

We still need to "find" GF(p) inside F. Consider the series 1, 1+1, 1+1+1,... Since F is finite, eventually some element must appear (say at the i^th and j^th places, i<j). So 1+1+..+1 j-i times is necessarily 0. Let p be the smallest natural number for which 1+1+...+1 (p times) = 0. It follows that if we write "n" for the element 1+1+...+1 (n times), then addition and multiplication on elements of this form are identical to the operations modulo p. p must be prime, else p=ab and then (in the field F) a+a+...+a (b times) = 0, which implies that 1+1+...+1 (b times) = 0, with b

QED

We haven't shown that there actually exists a field with pⁿ elements for every prime p and natural n, but this is true too. One representation of this field is to take polynomials in GF(p), modulo an irreducible polynomial of degree n.