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.



perfect fields in positive characteristic
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.


