THEOREM: Let $\displaystyle E$ be a field of $\displaystyle p^n$ elements contained in an algebraic closure $\displaystyle \bar{\mathbb{Z}_p}$ of $\displaystyle \mathbb{Z}_p$. The elements of $\displaystyle E$ are precisely the zeros in $\displaystyle \bar{\mathbb{Z}_p}$ of the polynomial $\displaystyle x^{p^n}-x$ in $\displaystyle \mathbb{Z}_p[x]$.

PROOF: The set $\displaystyle E^*$ of non-zero elements of $\displaystyle E$ forms a multiplicative group of order $\displaystyle p^n-1$ under the field multiplication. For $\displaystyle \alpha \in E^*$, the order of $\displaystyle \alpha$ in this group divides $\displaystyle |E^*|=p^n-1$. Thus for $\displaystyle \alpha \in E^*$ we have $\displaystyle \alpha^{p^n-1}=1, $ so $\displaystyle \alpha^{p^n}=\alpha$. Therefore every element of $\displaystyle E$ is a zero of $\displaystyle x^{p^n}-x$. Since $\displaystyle x^{p^n}-x$ can have atmost $\displaystyle n$ zeros, we see that $\displaystyle E$ contains precisely the zeros of $\displaystyle x^{p^n}-x$ in $\displaystyle \bar{\mathbb{Z}_p}$.

QUERY: Where have we used thealgebraic closurenessof $\displaystyle \bar{\mathbb{Z}_p}$. Wouldn't the proof still work if instead of $\displaystyle \bar{\mathbb{Z}_p}$ we had any field $\displaystyle K$ which is an extension field of the field $\displaystyle \mathbb{Z}_p$ containing the field $\displaystyle E$.

May be i am missing the obvious but can't figure it out.