IDA: Interactive Document on Algebra

Here is a way of finding isomorphisms between certain fields.

Proposition

Let K be a subfield of L and let f K[X] be an irreducible polynomial. If x, y are two roots of f in L, then there is an isomorphism h : K(x) -> K(y) with h(x) = y.

On the other hand, each isomorphism K(x) - > K(y) which fixes K elementwise, is determined by the image of x. Hence we can determine automorphism groups of finite fields.

Theorem

Let p be a prime number and q = p^a a power of p. If K is a finite field of order q, then Aut(K) is a cyclic group of order a generated by the map x -> x^p.