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It seems like this particular theorem is always stated in a way that's slightly hard to interpret. Let $S$ be some finite set of primes containing all the primes of $K$ ramifying in $L/K$. Then the statement is that the map

$$\psi:I^S_K\to Gal(L/K)$$

admits a modulus $\mathfrak{m}$ s.t. $S(\mathfrak{m})\subset S$ and we get a factorization

$$\psi:I^S_K\to I^{\mathfrak{m}}_K/i(K_{\mathfrak{m},1})\to Gal(L/K)$$

I was wondering if we always know precisely what the kernels are for both maps in the factorization below? Is it dependent on $\mathfrak{m}$? I've seen

$$i(K_{\mathfrak{m},1})\cdot Nm_{L/K}(I^{\mathfrak{m}'}_L)$$

show up in some places where $\mathfrak{m}'$ is the modulus consisting of the set of primes in $L$ lying above the primes in $\mathfrak{m}$. How does this norm thing fit in?

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Yes, one knows the kernels precisely; they are described by class field theory. The idelic formulation makes precise statements easier, though; have you seen the idelic point of view? If not, from what source are you getting your description? –  Matt E May 3 '11 at 0:35

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