normal basis theorem-proof finite fields

Question

I'm studying basis normal theorem from Prof. J. Milne's course notes. I've understood the proof for infinite fields but i don't get the proof for finite fields. I report below his proof.

Assume that $G$ is generated by an element $a_0$ of order $n$. The minimum polynomial of $a_0$ regarded as an endomorphism of the $F$-vector space $E$ is the monic polynomial in $F[x]$ of least degree such that $P(a_0)=0$ (as an endomorphism of $E$). It has the property that it divides every polynomial $Q(X)\in F[x]$ such that $Q(a_0)=0$. Since $a_0^n=1$, $P(x)$ divides $X^n -1$.

On the other hand, Dedekind’s theorem on the independence of characters (5.14) implies that $1,a_0,\ldots,a_0^{n-1}$ are linearly independent over $F$, and so $\deg P(x)\geq n$. We conclude that $P(x)=X^n-1$. Therefore, as an $F[x]$-module with $X$ acting as $a_0$, $E$ is isomorphic to $F[x]/(X^n-1)$.

For any generator $\alpha$ of $E$ as an $F[x]$-module, $\alpha,a_0(\alpha),\ldots,a_0^{n-1}(\alpha)$ is an $F$-basis for $E$.

In particular I'd like to understand what does he mean when he talks about minimum polynomial of $a_0$.

Thanks.

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What is it that you do not understand about the minimal polynomial of $a_0$ in particular? — awllower, Mar 14 '20 at 12:15
I recalled having answered a closely related question here. Does that answer your question? Actually this may be closer to what you need. It is essentially the same standard argument for the existence of a normal basis for a finite extension of finite fields. — Jyrki Lahtonen, Mar 14 '20 at 12:18
The difference between the argument I used and the one Milne is using is that the "standard" argument doesn't need independence of characters. That's because in the case of finite fields a counting argument can be used instead. Anyway, the argument Milne is using applies to any cyclic Galois extension (without the assumption that the fields are finite). — Jyrki Lahtonen, Mar 14 '20 at 12:29
And, to answer the question in your last sentence, the minimal polynomial of $a_0$ refers to its minimal polynomial as an $F$-linear endomorphism on a finite dimensional space. The minimal polynomial of its matrix (w.r.t. some basis), if you prefer to think about it that way. — Jyrki Lahtonen, Mar 14 '20 at 12:32

normal basis theorem-proof finite fields

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