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Question: Let $F$ be a field and fix its algebraic closure $\overline{F}$. Let $L_p$ be the compositum of all Galois $p$-extensions of $F$ in $\overline{F}$ where $p$ is a prime.

Show that any element in $L_p$ is contained in some $p$-extension of $F$.

My attempt: Let me choose $\alpha \in L_p$. I am trying to construct a $p$-extension of $F$ containing $\alpha$.

My initial thought is to consider the minimal polynomial $f_\alpha$ of $\alpha$ over $F$ and if I can show that the splitting field of $f_\alpha$ is a $p$-power extension of $F$, then I am done. However I have difficulty showing this.

Any help would be greatly appreciated.

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Vocabulary : a $p$-extension of $F$ is a finite Galois extension of degree a power of $p$. Key elementary lemma : the composite of two $p$-extensions of $F$ is again a $p$-extension. It follows that the composite $L_p$ of all finite $p$-extensions of $F$ is Galois open (because of maximality), but possibly of infinite degree. By infinite Galois theory, $G:=Gal(L_p/F)$ is a pro-p-group, i.e. a projective limit of finite $p$-groups, which is naturally a topological group with a fundamental system of open neighbourhoods of $1$ consisting of normal subgroups of finite index. Because of the Galois correspondance, for any $\alpha \in L_p$, the normal closure $L$ of $F(\alpha)$ is fixed by a normal closed subgroup $H $ of $G$ (which is open because it is of finite index). Almost by definition, $Gal(L/F)=G/H$ is a (finite) $p$-group.

A convenient reference is Cassels-Fröhlich, chap. V.

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