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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.

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  • $\begingroup$ Yes, it is necessary. Just observe that such $\;E\;$ is a $\;\Bbb F_p$-vector space of dimension $\;n\;$ ... $\endgroup$ – DonAntonio Apr 27 '14 at 4:39
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Theorem 2.1 here would be of interest to you. Also note that any field with $p$ elements is isomorphic to $\mathbb{Z}_p$.

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