Show that x -> x^p is an automorphism in K

Let K be a finite field of characteristic p > 0. Show that s(x) = x^p is an automorphism in K.

It is clear in case K is a prime field, because then s(x) is just an identity. If K is not prime, s(x) is identity on its prime subfield. But how to show the automorphism of s() for the rest of the field, when it's not prime?