Prime Power Mapping on Galois Field is Automorphism

Theorem
Let $\F$ be a Galois field whose zero is $0_\F$ and whose characteristic is $p$.

Let $\sigma: \F \to \F$ be defined as:
 * $\forall x \in \F: \map \sigma x = x^p$

Then $\sigma$ is an automorphism of $\F$.

Proof
Let $x, y \in \F$.

Then:

and:

Thus it has been demonstrated that $\sigma$ is a homomorphism.

Then we have:

From Kernel is Trivial iff Monomorphism, $\sigma$ is a monomorphism.

That is, $\sigma$ is an injection.

Then from Injection from Finite Set to Itself is Surjection, $\sigma$ is a surjection.

Thus $\sigma$ is a bijective homomorphism to itself.

The result follows by definition of automorphism.