p cohomological dimension of a profinite group

Question

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?

Answer 1

Assume that $p \neq 2$ or that $k$ is totally imaginary. Then the cohomological $p$-dimension of $G=\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$ is $\leq 2$ and $=1$ if and only if $H^2(U,\mathbb{Z}/p)=0$ for every open subgroup $U$ of $G$. Since the weak Leopoldt conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension, we have $H^2(U,\mathbb{Q}_p/\mathbb{Z}_p)=0$ and the vanishing of $H^2(U,\mathbb{Z}/p)$ is equivalent to the divisibility of $H^1(U,\mathbb{Q}_p/\mathbb{Z}_p)$. This, however, is equivalent to $\mu=0$ in the sense of Iwasawa theory. Hence the answer to the question on the cohomological $p$-dimension of $G$ is: $\le 2$ and $=1$ if the conjecture $\mu=0$ holds for the cyclotomic $\mathbb{Z}_p$-extension of every finite subextension of $k_S/k$.