Proof

Let z K be a primitive element of K (its existence is guaranteed by a previous theorem). An automorphism of K is determined by the image of z. Let f be the minimal polynomial of z over Z/pZ. Then f has degree a. Let g be an automorphism of K. Then g(z) is a zero of f. Hence there are at most a possibilities for g(z). On the other hand, all a possibilities occur:

z, z^p, z^p2, ..., z^pa-1

So Aut(K) is a cyclic group of order a generated by the automorphism sending z to z^p.