Power in finite field

Question

Does the following statement hold true for any finite field?

$$a^p\equiv a \qquad(\mathbb{Z_p})$$

I have tought at it this way: all numbers in $\mathbb{Z_p}$ are $\in \{0,\mathbb{Z_p}\}$ and $p*a< a^p< p^p=p\iff a< a^p< p$

I still miss something

Answer 1

Each element $a$ of a field $F$ of cardinality $q$ is a root of $$X^q -X,$$ so $a^q = a$ in $F$.

Note though that you need the cardinality of the field (not its characteristic), so the $q$ might not be a prime number.

Answer 2

The fact that this is true for any field of prime order is known as Fermat's little theorem. For other finite fields, however, this need not hold.