# A galois group is topologically finitely generated or finitely generated as a $\mathbb{Z}_p$-module

The basic set up is the following: Let $K$ be a number field. let $p$ be an odd prime. Let $\Sigma_p$ be the set of primes of $K$ lying above $p$. Let $M$ be the composite of all finite $p$-extensions of $K$ which are unramified outside $\Sigma_p$. $M^{ab}$ is the maximal abelian extension of $K$ contained in $M$. Let $\Gamma=Gal(M/K)$, then $\Gamma^{ab}\cong Gal(M^{ab}/K)$.

I was reading a paper where the author writes that it is well known that $\Gamma^{ab}$ is (topologically) finitely generated, which is the same thing as saying that $\Gamma^{ab}$ is finitely generated as a $\mathbb{Z}_p$-module. Then the author starts writing the notation $rank_{\mathbb{Z}_p}(\Gamma^{ab})$.

It will be very useful if someone gives me the reference of the fact that $\Gamma^{ab}$ is finitely generated.

• I don't have the book at home so I can't give a precise reference, but this fact is essentially proved in Washington's book on cyclotomic fields, in proving a bound on the number of "independent" $\mathbf{Z}_p$-extensions of a number field. The proof is based on the idelic description of global class field theory and the key fact is that the group of principal units in a $p$-adic field is a finitely generated $\mathbf{Z}_p$-module (which is the same thing as a topologically finitely generated abelian pro-$p$ group). – Keenan Kidwell May 31 '15 at 13:03