Assume that a field F has pn elements Show that the map F ri

Assume that a field F has p^n elements. Show that the map: F rightarrow F defined by (x) = x^p is an automorphism of F. is called the Frobenius automorphism. Determine the fixed field of.

Solution

Let us denote F by G.

is a homomorphism, so we simply need to prove that it is one-to-one and onto.

To see that is one-to-one, suppose that a ker , i.e., that (a) = e.

Then aⁿ = e, so o(a) must divide n. But o(a) also divides |G| by Lagrange’s theorem, and since gcd(n, |G|) = 1, we must have o(a) = 1. Therefore, a = e, so ker = {e}, and is one-to-one.

Since G is a finite field, any one-to-one map from G to G is necessarily onto. Therefore, is an automorphism of G.