I've arrived at a Theorem in text that I'm confused about:
Note: My question below is about the statement of this theorem, not about a proof for it. (The proof is supplied in the text)
Theorem: Let $E$ be a field of $p^{n}$ elements contained in an algebraic closure $\tilde{\mathbb{Z}_{p}}$ of $\mathbb{Z}_{p}$. The elements of $E$ are precisely the zeros in $\tilde{\mathbb{Z}_{p}}$ of the polynomial $x^{p^{n}} - x$ in $\mathbb{Z}_{p}[x]$.
The first line startles me somewhat. So far in this book we have never considered the algebraic closure of any structure which wasn't a field. And for $\mathbb{Z}_{p}$ to be a field, we must have that $p$ is prime. This is not given in the theorem, and there was no blanket statement at the beginning of the section as there sometimes is.
My Question: Have I missed some key fact regarding the orders of a finite field needing to be prime powers?
To give perspective to my background and where this chapter fits into development, the purpose of the chapter I am reading is to build the Galois Field of order $p^{n}$, with which I am not yet familiar.