I make self-study in class field theory and I want to prove the following popular fact:
Given a modulus $\mathfrak{m}$ of a number field $K$, the map $L\mapsto ker (\phi_{L/K,\mathfrak{m}}$) is an inclusion-reversing bijection between the set of finite abelian extensions of $K$ that admit $\mathfrak{m}$ and the set of congruence subgroups for $\mathfrak{m}$.
An extension $L/K$ admits $\mathfrak{m}$ iff $\mathfrak{m}$ is divisible by all primes that ramify in $L$ and $ker(\phi_{L/K,\mathfrak{m}})$ is congruence subgroup where $\phi_{L/K,\mathfrak{m}}$ is the Artin map.
As a result of Takagi existence theorem, the map is surjective. But I have problem about how to show it is injective.
Your problem, in the equivalent form stated by @mercio, precedes class field theory. The THEOREM says: Let $L_1 , L_2$ be finite Galois extensions of a number field $K$ , and $S_1, S_2$ be the sets of primes which split completely from $K$ to $L_1 , L_2$ resp. Then $S_1 \subset S_2$ (with finitely many exceptions) if and only if $L_2 \subset L_1$. Thus $S_1 = S_2$ if and only if $L_1 = L_2$. You can find this as an exercise around the Tchebotarev density theorem in Cassels-Fröhlich (ex. 6, p. 362).