Abelian subgroups of pro-𝑝 Galois groups

Engler, Antonio; Koenigsmann, Jochen

doi:10.1090/S0002-9947-98-02063-7

Abelian subgroups of pro-$p$ Galois groups
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by Antonio José Engler and Jochen Koenigsmann

Trans. Amer. Math. Soc. 350 (1998), 2473-2485

DOI: https://doi.org/10.1090/S0002-9947-98-02063-7

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Abstract:

It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois $p$-extension of a field $F$ (where $p$ is an odd prime) arise from $p$-henselian valuations with non-$p$-divisible value group, provided $\# (\dot {F}/\dot {F}^{p})\geq p^{2}$ and $F$ contains a primitive $p$-th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.

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