Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group

Zakariae Bouazzaoui; Abdelaziz El Habibi

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arXiv · 2112.05027

Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group

Zakariae Bouazzaoui·Abdelaziz El Habibi

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Abstract

Let $k$ be an imaginary quadratic field and $p$ an odd prime number such that the $p$-rank of the class group of $k$ is one. Let $S$ be a finite set of places of $k$ distinct from $p$-adic places. We give sufficient conditions for the Galois group $G_S$, of the maximal pro-$p$-extension of $k$ which is unramified outside $S$, to be \textit{mild}, hence of cohomological dimension $2$.

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Zakariae Bouazzaoui, Abdelaziz El Habibi. 2021-12-09. Mild pro-$p$-groups and $p$-extensions of imaginary quadratic fields with non-trivial $p$-class group. https://arxiv.org/abs/2112.05027

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