Proof of Fermat's Little theorem using Field theory

Question

I wanted to conclude Fermat's Little theorem form below fact.I had proved following facts but unable to conclude result

Let F be finite field with char $p\neq 0$ then $f:F\to F$ such that $f(x)=x^p$ is automorphism

Also I know that any field automorphism is identity on prime subfield

How can above 2 facts leads to proof that

For $a\in \mathbb Z$

$a^p-a=0\mod p$

Any Help Hint will be appreciated

Answer 1

Well, ${\Bbb Z}_p$ is the prime field of characteristic $p$. The mapping $x\mapsto x^p$ is an automorphism of ${\Bbb Z}_p$ and the prime field is fixed under an automorphism. It follows that $x^p=x$ for each element $x$ of ${\Bbb Z}_p$, which is equivalent to $x^p\equiv x\mod p$.