# \$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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