Notes on class field theory

Recall that the ideal class group $\mathrm{Cl}(K)$ of $K$ is defined as the group $J_K$ of fractional ideals modulo the subgroup $P_K$ of principal fractional ideals. Let $\mathfrak{m}$ be a formal product of places of $K\text{;}$ you may regard such a beast as an ordinary integral ideal together with a nonnegative coefficient for each infinite place.

Let $L/K$ be a (finite) abelian extension of number fields. For each prime $\mathfrak{p}$ of $K$ that does not ramify in $L\text{,}$ let $\mathfrak{q}$ be a prime of $L$ above $K\text{,}$ and put $\kappa ={\mathfrak{o}}_K/\mathfrak{p}$ and $\lambda ={\mathfrak{o}}_L/\mathfrak{q}\text{.}$ Then the residue field extension $\lambda /\kappa$ is an extension of finite fields, so it has a canonical generator $\sigma \text{,}$ the Frobenius automorphism, which acts by raising to the $q$-th power. (Here $q=\mathrm{Norm}(\mathfrak{p})=\mathrm{\#}\kappa$ is the absolute norm of $\mathfrak{p}\text{.}$) Since $\mathfrak{p}$ does not ramify, the decomposition group $G_{\mathfrak{q}}$ is isomorphic to $\mathrm{Gal}(\lambda /\kappa )\text{,}$ so we get a canonical element of $G_{\mathfrak{q}}\text{,}$ called the Frobenius of $\mathfrak{q}\text{.}$ In general, replacing $\mathfrak{q}$ by ${\mathfrak{q}}^{\tau }$ for some $\tau \in \mathrm{Gal}(L/K)$ conjugates both the decomposition group and the Frobenius by $\tau \text{;}$ since $L/K$ is abelian in our case, that conjugation has no effect. Thus we may speak of “the Frobenius of $\mathfrak{p}$” without ambiguity.

Define the conductor of $L/K$ to be the smallest formal product $\mathfrak{m}$ for which the conclusion of Theorem 2.2.6 holds. We say $L/K$ is the ray class field corresponding to the product $\mathfrak{m}$ if $L/K$ has conductor dividing $\mathfrak{m}$ and the map $J_K^{\mathfrak{m}}/P_K^{\mathfrak{m}}\to \mathrm{Gal}(L/K)$ is an isomorphism.