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…)

It is now time to begin the final descent towards the Artin reciprocity law, which states that for an abelian extension ${L/k}$, there is an isomorphism

$\displaystyle J_k/k^* NJ_L \simeq G(L/k).$

We will actually prove the Artin reciprocity law in the idealic form, because we have only defined the Artin map on idelas. In particular, we will show that if ${\mathfrak{c}}$ is a suitable cycle in ${k}$, then the Artin map induces an isomorphism

$\displaystyle I(\mathfrak{c}) / P_{\mathfrak{c}} N(\mathfrak{c}) \rightarrow G(L/k).$

The proof is a bit strange; as some have said, the theorems of class field theory are true because they could not be otherwise. In fact, the approach I will take (which follows Lang’s Algebraic Number Theory, in turn following Emil Artin himself).

So, first of all, we know that there is a map ${I(\mathfrak{c}) \rightarrow G(L/k)}$ via the Artin symbol, and we know that it vanishes on ${N(\mathfrak{c})}$. It is also necessarily surjective (a consequence of the first inequality). We don’t know that it factors through ${P_{\mathfrak{c}}}$, however.

Once we prove that ${P_{\mathfrak{c}}}$ (for a suitable ${\mathfrak{c}}$) is in the kernel, then we see that the Artin map actually factors through this norm class group. By the second inequality, the norm class group has order at most that of ${G(L/k)}$, which implies that the map must be an isomorphism, since it is surjective.

In particular, we will prove that there is a conductor for the Artin symbol. If ${x}$ is sufficieintly close to 1 at a large set of primes, then the ideal ${(x)}$ has trivial Artin symbol. This is what we need to prove.

Our strategy will be as follows. We will first analyze the situation for cyclotomic fields, which is much simpler. Then we will use some number theory to reduce the general abelian case to the cyclotomic case (in a kind of similar manner as we reduced the second inequality to the Kummer case). Putting all this together will lead to the reciprocity law.
(more…)

Class field theory is about the abelian extensions of a number field ${K}$. Actually, this is strictly speaking global class field theory (there is an analog for abelian extensions of local fields), and there is a similar theory for function fields of transcendence degree 1 over finite fields, but we shall not deal with it.

Let us, however, consider the situation for local fields—which we will later investigate more—as follows. Suppose ${k}$ is a local field and ${L}$ an unramified extension. Then the Galois group ${L/k}$ is isomorphic to the Galois group of the residue field extension, i.e. is cyclic of order ${f}$ and generated by the Frobenius. But I claim that the group ${k^*/NL^*}$ is the same. Indeed, ${NU_L = U_K}$ by a basic theorem about local fields that we will prove using abstract nonsense later (but can also be easily proved using successive approximation and facts about finite fields). So ${k^*/NL^*}$ is cyclic, generated by a uniformizer of ${k}$, which has order ${f}$ in this group. Thus we get an isomorphism

$\displaystyle k^*/NL^* \simeq G(L/k)$

sending a uniformizer to the Frobenius. (more…)