Class field theory

Remarks

The main theorems are as follows. Each congruence subgroup $H$ has a class field $L$, unique inside a fixed algebraic closure. This $L/K$ is abelian and unramified at the places not dividing $\mathfrak{m}$. The Artin map gives $\Gal(L/K) \cong I_K^{\mathfrak{m}}/H$. Every finite abelian extension of $K$ is the class field of some congruence subgroup. The ray class group is finite, and its order is a multiple of the class number.

Example: $K = \Q$, $\mathfrak{m} = n\infty$ with $n \ge 3$, and $H = P_{\Q}^{\mathfrak{m}}$. For a prime $p \nmid n$, the ideal $(p)$ has the unique positive generator $p$, so $(p) \in H$ if and only if $p \equiv 1 \pmod n$. This is exactly the condition for $p$ to split completely in the cyclotomic field $\Q(\zeta_n)$, so $\Q(\zeta_n)$ is the class field of $H$.

Non-example: for $\mathfrak{m} = 5$ without the infinite place, $(p) = (-p)$ lies in $P_{\Q}^{\mathfrak{m}}$ when $p \equiv \pm 1 \pmod 5$. So $\Q(\zeta_5)$ is not the class field of $P_{\Q}^{\mathfrak{m}}$: the prime $19 \equiv -1 \pmod 5$ gives $(19) \in P_{\Q}^{\mathfrak{m}}$, but $19$ does not split completely in $\Q(\zeta_5)$. The class field of this group is the real subfield $\Q(\zeta_5 + \zeta_5^{-1}) = \Q(\sqrt{5})$.