I would like to know what is the $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$. Here $S$ is a finite set of primes containing $p$ and the Archimedean primes and $\mathbb{Q}_{cyc}$ is the cyclotomic $\mathbb{Z}_p$-extension.

I've been looking for a reference and have been unable to find it.

The $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q})$ is 2 and Proposition 3.3.5 (Neukirch-Schmidt-Wingberg, second edition corrected) implies that the $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$ is less than or equal to 2.

It also seems to follow from a result of Neukirch (see Theorem 10.5.6 NSW) that $\textrm{Gal}(\mathbb{Q}_S(p)/\mathbb{Q}_{cyc})$ has $p$-cohomological dimension less than equal to 1, where $\mathbb{Q}_S(p)/\mathbb{Q}$ is the maximal $p$-quotient..

But I don't know what can be said about the extension $\mathbb{Q}_S/\mathbb{Q}_S(p)$. Is this extension prime to $p$?

In fact, suppose $p$ is an odd prime and $S$ is any finite set containing the primes above $p$ and the Archimedean primes. Does there exist any number field $K$ such that $\textrm{Gal}(K_S/ K_{cyc})$ has $p$-cohomological dimension 1?


This question had a bounty worth +50 reputation from debanjana that ended 21 hours ago. Grace period ends in 2 hours

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  • $\begingroup$ What is $\mathbb{Q}_S?$ $\endgroup$ – Dedalus Sep 6 at 15:35
  • 3
    $\begingroup$ The maximal unramified outside of $S$-extension? $\endgroup$ – Dedalus Sep 6 at 15:35
  • $\begingroup$ Yes, $\mathbb{Q}_S$ is the maximal unramified outside $S$-extension.. $\endgroup$ – debanjana Sep 6 at 15:38

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