It is now time to prove the reciprocity law, the primary result in class field theory.  I know I haven’t posted on this topic in a little while, so new readers (if they don’t already know this material) may want to review the strategy of the proof and the meaning of the Artin lemma (which is useful in reducing this to the cyclotomic case).

1. The cyclic reciprocity law

Well, I’ve already stated it before multiple times, but here it is:

Theorem 1 (Reciprocity law, cyclic case) Let ${L/k}$ be a cyclic extension of number fields of degree ${n}$. Then the reciprocity law holds for ${L/k}$: there is an admissible cycle ${\mathfrak{c}}$ such that the kernel of the map ${I(\mathfrak{c}) \rightarrow G(L/k)}$ is ${P_{\mathfrak{c}} N(\mathfrak{c})}$, and the Artin map consequently induces an isomorphism$\displaystyle J_k/k^* NJ_L \simeq I(c)/P_{\mathfrak{c}} N(\mathfrak{c}) \simeq G(L/k).$

The proof of this theorem is a little sly and devious.

Recall that, for any admissible cycle ${\mathfrak{c}}$, we have

$\displaystyle (I(\mathfrak{c}): P_{\mathfrak{c}} N(\mathfrak{c})) = n$

by the conjunction of the first and second inequalities, and the Artin map ${I(\mathfrak{c}) \rightarrow G(L/k)}$ is surjective. If we prove that the kernel of the Artin map is contained in ${P_{\mathfrak{c}} N(\mathfrak{c})}$, then we’ll be done by the obvious count.

This is what we shall do. (more…)