# p cohomological dimension of a profinite group

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

Looking for an answer drawing from credible and/or official sources.

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