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Prove that finite separable normal extension $\mathbb{F}$ of field $\mathbb{k}$, $\operatorname{char}(\mathbb{k})=p > 0$ has Galois abelian group $\operatorname{Gal}(\mathbb{F}/\mathbb{k})$ with period p if and only if $\mathbb{F} = \mathbb{k}(\alpha_1, \dots, \alpha_n)$, $\alpha_i^p - \alpha_i - a_i = 0$, $a_i \in \mathbb{k}^*$.

I have just started learning Galois theory, any advice will be highly appreciated. Thanks in advance.

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    $\begingroup$ In this oldish answer I give a (standard text book) proof for the fact that if $L/K$ is a cyclic extension of degree $p=\text{char} L$, then $L=K(\alpha)$ for some $\alpha$ such that $\alpha^p-\alpha=z\in K$. This is a standard Lemma of Artin-Schreier theory. Does that help? $\endgroup$ – Jyrki Lahtonen Jan 29 '16 at 16:06

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