Even more about Galois Fields Here's one interesting question that I found in my textbook. But first, here's some background information:

Let $p$ be a prime number, and $P(x)$ be an irreducible polynomial over $\mathbb{Z_p}$. Let deg $P = v$. Let $F = GF(p,P(x))$ be the corresponding Galois field and notice that $\mathbb{Z_p}$ is a subfield of $F$.

Here's is what I need to show:

Show that, for any element $a$ of $F$, $a^p = a$ if anf only if $a$ belongs to $\mathbb{Z_p}$.

First of all, I don't like how this question if phrased. I prefer to use the notation $\mathbb Z/p\mathbb Z$ instead of $GF(p,P(x))$. I've been collaborating with the people in my math class on how to solve this problem, but haven't gotten far. I defined $P(x)$ to be a non-zero polynomial such that $$P(x) = a_0 + a_1x+ a_2x^2 + ... + a_{v-1}x^{v-1}$$
$\Rightarrow$ Rough sketch is the use Fermat's Little Theorem.
$\Leftarrow$ I believe that the fact that $\mathbb{Z_p}$ is a subfield of $F$ should be considered here.
That's as far as I've gotten. 
 A: The map $a \mapsto a^p$ is important in the study of finite fields.  For more information, you can Google "Frobenius endomorphism".  The reason this is interesting is that composing $a \mapsto a^p$ with itself over and over will generate the entire automorphism group of the finite field of order $p^n$.  Indeed, Galois groups of finite fields are cyclic.  Anyways...
You have the right idea about the forward direction.  If we think group-theoretically, $\mathbb{Z}_p^\times$ is a group under multiplication of order $p-1$.  Therefore, $a^{p-1} = 1$ as a consequence of Lagrange's theorem, which gives $a^p = a$ (this is essentially just a re-proof of FLT).  Of course this is also true for $0$, so we have $a^p = a$ for all $a \in \mathbb{Z}_p$.
For the converse, Crostul in the comments has the right idea.  Consider the polynomial $f(x) = x^p - x$.  By the above, all of $\mathbb{Z}_p$ satisfies this polynomial, and the number of roots of a polynomial over a field (counting multiplicity) is always equal to the degree of that polynomial.
