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 :-)