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I am trying to answer this exercise from Hungerford’s Algebra under Cyclic extensions section.

For the forward direction, assume $K$ has a cyclic extension of degree $p$. Then $\text{Aut}_K F$ is cyclic. How do I choose an element of $K_p$ that is not in $K$? I am thinking that I should consider the Frobenius Automorphism and the Fundamental Theorem of Galois Theory, but I’m not sure how.

For the backward direction, suppose $K_p \neq K$. I think $F$ can be chosen such that it is a splitting field of some polynomial, or I need to show that $\text{Aut}_K F$ is generated by the Frobenius automorphism. But I’m not sure how to use the assumption.

Please help me with this one. Thank you so much!

Mashed Potato
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    A starting point of Artin-Schreier theory of abelian $p$-extensions of characteristic $p$ fields is the result that if $F/K$ is cyclic of degree $p$, then $F=K(\alpha)$, where $$\alpha^p-\alpha\in K\setminus {u^p-u\mid u\in K}.$$ See my old answer for an on-site proof. Conversely, if $\beta\in K$ is not of the form $u^p-u$, then it follows that the splitting field $F$ of $x^p-x-\beta$ is cyclic of degree $p$. – Jyrki Lahtonen Sep 22 '23 at 03:49
  • But i'm not sure that piece really answers your question. We cannot always use the Frobenius automorphism here, for typically $K$ is not fixed by it where as Galois theory calls for $K$-automorphisms of $F$. – Jyrki Lahtonen Sep 22 '23 at 03:52
  • Another sentence caught my eye. If by $K_p$ you mean the set of elements of the form $u^p-u$, $u\in K$, then surely always $K_p\subseteq K$. The real question is whether all the elements of $K$ can be written in that form, and the conclusion of this exercise is that $K$ has cyclic degree $p$ extensions if and only if $K_p$ is a proper subset of $K$. For example, should $K$ be algebraically closed, then trivially $K_p=K$ as no finite extensions can exist in that case. – Jyrki Lahtonen Sep 22 '23 at 03:55
  • In other words, you may be trying to unravel this from the wrong direction? Not unnatural when diving into previously unknown waters :-) But please clarify further. – Jyrki Lahtonen Sep 22 '23 at 03:57
  • Undoubtedly earlier in the book there was a result (a theorem or an exercise) showing that if $K$ A) has characteristic $\neq p$, and B) there exists a root of unity $\zeta$ of order $p$ in $K$, then a cyclic degree $p$ extension $F/K$ is of the form $F=K(\alpha)$, where $\alpha^p=z\in K$, and $z$ is not of the form $u^p, u\in K$. The point of this exercise really is to see how that result changes, when we are in characteristic $p$. One key ingredient is that instead of $p$th powers like $u^p$ we need to exclude elements of the form $u^p-u$ when describing the extensions. – Jyrki Lahtonen Sep 22 '23 at 04:03
  • Sorry about the longish train of comments as opposed to an answer. 1) we have similar threads on the site already, so an answer would likely be a duplicate, 2) it is still not entirely clear to me what exactly is the gist of your question (as we should not simply solve the exercise for you as per site policies). – Jyrki Lahtonen Sep 22 '23 at 04:05
  • @JyrkiLahtonen I think I am kinda getting it now. I am looking at the wrong things. Thank you for pushing me into this direction. – Mashed Potato Sep 22 '23 at 07:07

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