$p$ cohomological dimension of profinite groups

by debanjana   Last Updated September 11, 2019 17:20 PM

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?

Here $K_S$ is the maximal unramified extension of $K$ outside $S$ and $K_cyc$ is the cyclotomic $\mathbb{Z}_p$-extension of $K$.

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