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I'm reading a note about structures of finite fields. In chapter 9 the author aims at proving that two finite fields $F,F'$ with the same number ($q$) of elements are necessarily isomorphic. I'm particularly interested in his proof because his proof seems to avoid the usage of stuff like splitting field.

As far as I can see, the idea of his proof is to find two generators $\pi,\pi'$ of $F^*,F'^*$ respectively, and the isomorphism naturally follows. But I do not quite understand some of his arguments.

Now I quote the start of his proof here:

Choose a primitive root $\pi\in F$. Let its minimal polynomial be $m(x)$. Then $m(x) \mid x^q-x$. Now go across to $F'$. Since

$$ x^q-x=\prod_{a'\in F'}(x-a'), $$

$m(x)$ must factor completely in $F'$. (... the rest of the proof omitted)

Question: What does the author mean by 'go across to $F'$'? I really don't know how $m(x)\in F[x]$ can be directly regarded as a polynomial of $F'[x]$. Is his proof correct?

I also wonder if there're any 'elementary' proof of this uniqueness. I learnt the concept of splitting fields in an algebra course, but I want to explain the theorem and its proof to my friends taking a cryptography course with 'plainer' words.

strategyproof
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    Without looking, $m(x)$ is a polynomial with coefficients in the prime field $\Bbb{F}_p$. As is the polynomial $x^q-x$ ($q=p^n$, $p$ a prime). Both $F$ and $F'$ share that same prime field, so divisibility over $\Bbb{F}_p$ applies to both of them. BTW, 2 years ago I faced the same challenge to try and prove uniqueness up to isomorphism without using the concept of splitting fields, and I came up with this same (IMO natual) idea :-) – Jyrki Lahtonen Dec 13 '20 at 16:32
  • Basically $\Bbb{F}_p[x]$ is a shared subring of both $F[x]$ and $F'[x]$, so any element of $\Bbb{F}_p[x]$, such as $m(x)$, can be unambiguously seen as an element of either of the bigger rings. – Jyrki Lahtonen Dec 13 '20 at 16:33

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