# A finite field GF($p^n$) of $p^n$ elements exists for every prime power $p^n$.

I am stucked to prove :

A finite field GF($p^n$) of $p^n$ elements exists for every prime power $p^n$.

In my book Fraleigh P303 ,

we think $\overline{Z_p}$ be an algebraic closure of ${Z_p}$ and let $K$ be the subset of $\overline{Z_p}$ consisting of all zeros of $x^{p^n} -x$ in $\overline{Z_p}$.

Then it seeme like $K$ has a prime characteristic $p$, since we use lemma when a field of prime characteristic $p$.

I want to understand why $K$ has a prime characteristic.

Thank you

Well, $p\cdot 1=0$ in $\mathbb{Z}_p$ and hence also in $\overline{\mathbb{Z}_p}$, and hence also in $K$. More generally if $k$ is a field, then any subfield or field extension has the same characteristic.