$K$ is a field containing $\mathbb{Z}_p$, then $\alpha\in K$ and $\alpha^p=\alpha$ implies $\alpha\in\mathbb{Z}_p$

Question

suppose $K$ is a field containing $\mathbb{Z}_p$, then $\alpha\in K$ and $\alpha^p=\alpha$ implies $\alpha\in\mathbb{Z}_p$.

I've got a solution by algebraic algebra theory if $K$ is finite generated field of $\mathbb{Z}_p$, but still not sufficient to fulfill the prove when $K$ is infinite, i.e. $K=\mathbb{Z}_p(\pi)$. With the knowledge of elementary number theory, this prob. got difficult to me.

Answer 1

Over any field, the equation $X^p-X=0$ has degree $p$ and so at most $p$ solutions. But it has $p$ solutions in $\Bbb F_p$, so cannot have any more in any field containing $\Bbb F_p$.

Answer 2

$ x^p-x$ has degree $p$ so it has at most $p$ roots in $K$.

The set of this roots is $\mathbb{Z}_p$