Let K be a finite field. Recall that K has pn elements, for a prime p and a positive integer n and that there is exactly one such field up to isomorphism. We are going to compute the automorphisms of K and as a corollary we will use Galois theory to compute the subfields of K.

Theorem The automorphism group of K is cyclic of order n generated by the Frobenius endomorphism.

Corollary The subfields of K are exactly the fields with pr elements for r a divisor of n.

Proof of the theorem: First let's recall a few facts about K that we'll need in the proof. Firstly K has Zp as a subfield. Notice that if f:K-->K is an automorphism then it must be that f is the identity on Zp. For f(1)=1 and f(0)=0 and the elements of Zp are obtained by adding 1s together: 0,1,1+1,1+1+1,...,p-1 Thus any automorphism of K is actually a Zp-automorphism. It follows that the automorphism group of K is the same as Gal(K/Zp).

I claim that K is a Galois extension of Zp. Well it was shown in All finite fields are isomorphic to GF(p^n) that it is a splitting field of the polynomial h(x)=xpn-x over Zp. This polynomial is separable because its derivative is -1.

Thus we can apply Galois theory. It follows that Gal(K/Zp) has order n since this is the dimension of K as a vector space over Zp. Now consider the Frobenius endomorphism F:K-->K. We know that this is an automorphism of F. We have to show that it has order n. Since every element of K is a zero of h(x) it follows that Fn=1. Suppose that F has an order strictly smaller than this d, say. Then apd=a for each a in K. But this means that the elements of K are all zeroes of a polynomial with degree pd. In particular K can have at most pd elements, a contradiction. Thus the Frobenius has the correct order and the group is cyclic as required.

Proof of the corollary: From Galois theory we know that the subfields of K are exactly KG for the subgroups G of Gal(K/Zp)=<F>. and that [K:KG]=|G| (*).

The subroups of <F> are precisely <Fd> for d a divisor of n. By (*) the corresponding fixed field has dimension r over Zp and hence has pr elements.

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