# A question about finite fields with order $p^n$.

If we have a finite field $E$ with $p^n$ elements, where $p$ is a prime number. Is it necessary that it is a field extension of a subfield isomorphic to $\Bbb{Z_p}$, and consequently is of characteristic $p$?

I believe it is true that any finite field of characteristic $p$ has order $p^n$, and it is a field extension of a subfield isomorphic to $\Bbb{Z_p}$. However, I'm interested in the converse.

• Yes, it is necessary. Just observe that such $\;E\;$ is a $\;\Bbb F_p$-vector space of dimension $\;n\;$ ... – DonAntonio Apr 27 '14 at 4:39

Theorem 2.1 here would be of interest to you. Also note that any field with $p$ elements is isomorphic to $\mathbb{Z}_p$.