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Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:

  1. Let $L$ be a finite Galois extension of $k$. Is $\mathrm{Gal}(L/k)$ always cyclic?
  2. Let $L$ and $L'$ be two finite Galois extensions of $k$ of the same degree. Are $L$ and $L'$ always $k$-isomorphic fields?

The only example of such a field $k$ I have encountered is $\mathbb{F}_{q}(x,x^{p^{-1}},x^{p^{-2}},x^{p^{-3}},\ldots)$, where $q=p^d$ for some prime $p$, and the only finite Galois extensions I see are those of the form $\mathbb{F}_{q^e}(x,x^{p^{-1}},x^{p^{-2}},x^{p^{-3}},\ldots)$, which makes me expect a positive answer to both questions. (Edit: There are more (of course), see Emil Jeřábek's comment.)

Basically, I want to know whether Galois theory over infinite perfect fields in positive characteristic is as ''easy'' as over finite fields. In addition to an answer to the questions above, any properties supporting/contradicting this vague statement are apreciated.

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    $\begingroup$ Presumably, in (2), you want $L$ and $L'$ to be of the same degree? The property you are asking for seems closely related to being quasi-finite. $\endgroup$
    – LSpice
    Apr 13, 2021 at 12:37
  • $\begingroup$ Yes, I meant same degree, thanks. I edited it. $\endgroup$
    – JNS
    Apr 13, 2021 at 12:40
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    $\begingroup$ Your example field is in fact already a counterexample. For instance, where is the $l$-th root of $x$ for prime $l\ne p$? $\endgroup$ Apr 13, 2021 at 13:31
  • $\begingroup$ Galois theory of perfect fields is, in fact, as hard as the Galois theory of general fields: if $k$ is a file of characteristic $p$ then its finite separable extensions are in bijection with finite (necessarily separable) extensions of its perfection $k_{perf}=colim(k\xrightarrow{x\mapsto x^p} k\to \dots)$ $\endgroup$
    – SashaP
    Apr 14, 2021 at 2:20

1 Answer 1

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To produce an obvious counterexample to (1), fix a finite field $F$ of characteristic $p$, $n\ge 3$ and $L=\bigcup_m F[x_1^{p^{-m}},\dots,x_n^{p^{-m}}]$. This is a perfect field. The symmetric group $S_n$ acts by permuting the variables. Let $K$ be the fixed point field. Then $L$ is a finite Galois extension of $K$ with Galois group $S_n$.

A counterexample to (2), say with $p>2$: now the ground field is $L$. Let $F[u]$ be a quadratic extension of $F$. Then $L\subset L[u]$ and $L[x_1^{1/2}]$ are quadratic extensions of $L$, but are not isomorphic.

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    $\begingroup$ PS: the first construction, replacing $S_n$ with $G\subset S_n$ shows that for every prime $p$, every finite group is Galois group of a Galois extension of some perfect field of char. $p$. $\endgroup$
    – YCor
    Apr 15, 2021 at 6:56

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