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 :-)
Here are a few reasons why one shouldn't expect $Cl_S(k)$ to be related to $Gal(k_S / k)$, or even to $Gal(k_S^{\mathrm{ab}} / k)$ where $k_S^{\mathrm{ab}}$ is the maximal abelian extension unramified outside $S$.
Firstly, the Galois group of $k_S^{\mathrm{ab}} / k$ will generally be quite large; e.g. for $k = \mathbf{Q}$ the extension $k_S^{\mathrm{ab}} / k$ is infinite as soon as $S$ is non-empty. On the other hand $Cl_S(k)$ is always finite.
Moreover, the field $k_S^{\mathrm{ab}}$ gets bigger and bigger as you enlarge $S$, while the S-ideal class group gets smaller -- for $S$ sufficiently large it will be trivial.
If you are familiar with the isomorphism of global class field theory between $Gal(k^{\mathrm{ab}} / k)$ and the idele class group $\mathbf{A}_k / \overline{k^\times k_\infty^\circ}$, you can easily see what's going on. The group $Gal(k^{\mathrm{ab}}_S / k)$ corresponds to the quotient of the idele class group by the image of the subgroup of $\mathbf{A}_k^\times$ given by $$ \left( \prod_{v \notin S}\mathcal{O}_{k, v}^\times\right) \times \left(\prod_{v \in S} 1 \right).$$ The group $Cl_S(k)$ corresponds to the quotient by $$ \left( \prod_{v \notin S}\mathcal{O}_{k, v}^\times\right) \times \left(\prod_{v \in S} K_v^\times \right).$$ So these are very different beasts, unless $S$ is empty.