Checking Finite Field Automorphism

Question

My book contains the proof of the following claim:

Let $F$ be a finite field with characteristic p. Show that $F^p = F$

They construct a morphism $\phi: F \rightarrow F^p$ that sends $x \mapsto x^p$. The check this is a homomorphism, that it is one-to-one, and its onto. I followed most of the proof, except for checking surjection. They write one line:

Since F is finite, $\phi$ is onto.

Why is this true?

Answer 1

Since $F^p \subseteq F$, then $\#(F^p) \leq \#F$. Since $\phi: F \to F^p$ is injective, then $\#F \leq \#(F^p)$, so $\#(F^p) = \#F$. Since these are finite sets, then $F^p = F$. An injective map from a finite set to itself must also be surjective.

Answer 2

Here is a direct proof that $\phi$ is surjective.

Let $F$ have $q$ elements. Then $a^q = a$ for all $a \in F$ because of Lagrange's theorem applied to $F^{\times}$.

Since $q$ is a power of $p$, we have $q=tp$ and so $a=a^q=(a^t)^p \in F^p$.