Question about the modulus of a number field Milne defines the conductor of an abelian extension $L/K$ to be the smallest modulus $\mathfrak{m}$ s.t. the Artin map factors as
$$\psi_{L/K}:I_K^{\mathfrak{m}}\to \textrm{Cl}_\mathfrak{m}(K)\to \textrm{Gal}(L/K)$$
I'm trying to understand the class field theoretic proof of the Kronecker-Weber theorem. For the proof to work, one needs to show that given some conductor $\mathfrak{m}$ and its ray class field $L_\mathfrak{m}$, then the abelian extensions of $K$ contained in $L_\mathfrak{m}$ are precisely the fields s.t. their conductor divides $\mathfrak{m}$. Hence, one needs to prove (denoting the conductor of $L/K$ by $\mathfrak{f}(L/K)$) that
$$\mathfrak{f}(L/K)\mid \mathfrak{m}\Rightarrow L\subset L_\mathfrak{m}$$
Anyone know how to prove this? I've been looking at a few other books, but they seem to define things differently, so the proofs did not translate to Milne's definition, which is what I'm currently trying to understand.
EDIT: This question is about trying to fill in the details in Remark 3.8 on page 155 in his notes: http://www.jmilne.org/math/CourseNotes/CFT.pdf
 A: I've never read Milne's notes, so I don't know what you are giving yourself at this point.
One way to argue is as follows: If the conductor $\mathfrak f$ of $L$ over $K$
divides $\mathfrak m$, then all the primes that split completely in $L_{\mathfrak m}$ also split completely in $L$.  Consider now the compositum of $L$ and $L_{\mathfrak m}$; one  sees that the primes that split completely in this extension are precisely the primes that split completely in $L_{\mathfrak m}$.
But a standard consequence of Cebotarev density is: if $E_1$ and $E_2$ are two Galois extensions of the number field $K$ (in some fixed algebraic closure $\overline{K}$) and the sets of primes that splits completely in each of them coincide (perhaps away from a finite number of bad primes), then $E_1$ and $E_2$ are equal.  
This implies that $L_{\mathfrak m} L = L_{\mathfrak m}$, i.e. that $L \subset 
L_{\mathfrak m}$, as desired.
[In the context of abelian extensions there are other proofs that avoid 
Cebotarev; e.g. once you have the whole class field theory apparatus, the 
inclusion you ask about follows immediately.  But the above argument is a
stanard and important one, which gets at the heart of how splitting behaviour
can pin down a field --- after all, $L_{\mathfrak m}$ is defined a priori
by the splitting behaviour of primes, and it is the above result that shows
that this splitting behaviour suffices to pin down $L_{\mathfrak m}$ 
uniquely.]
