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. 



A galois group is topologically finitely generated or finitely generated as a $\mathbb{Z}_p$-module
See also [Neukirch-Schmidt-Wingberg], Cohomology of Number Fields, Theorem (11.1.2) and Proposition (10.3.20)(ii). You can find an online version at <https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html>


