TITLE: Interpretation of $S$-ideal class group
QUESTION [7 upvotes]: I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the maximal Galois extension which is unramified outside of $S$. 
Now the $S$-ideal class group $Cl_S(k)$ is stated to be "naturally isomorphic to the Galois group of the maximal abelian extension of $k$ inside $k_S$ in which all primes of $S$ split completely". By class field theory I know that $Cl(k)$ is isomorphic to the galois group of the maximal abelian unramified extension of $k$. But how does the relation to the splitting of primes work? Why is it not isomorphic to the Galois group of $k_S|k$?. 
Does somebody can help me or provide me with a reference?
Thank you a lot, Tom :-)

REPLY [3 votes]: Relationship between Cl_S(k) and the splitting of primes : Cl_S (k) is the quotient of Cl(k) by the subgroup generated by the classes of the prime ideals in S. If we identify by class field theory Cl(k) with the Galois group over k of the maximal abelian unramified extension of k (the so called Hilbert class field), then the aforementioned subgroup is isomorphic to the common decomposition subgroup of all the primes in S.