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…)

This is the lemma we shall use in the proof of the reciprocity law, to reduce the cyclic case to the cyclotomic case:

Lemma 4 (Artin) Let ${L/k}$ be a cyclic extension of degree ${n}$ and ${\mathfrak{p}}$ a prime of ${k}$ unramified in ${L}$. Then we can find a field ${E}$, a subextension of ${L(\zeta_m)}$, with ${E \cap L = k}$ such that in the lattice of fields

we have:

1. ${\mathfrak{p}}$ splits completely in ${E/k}$

2. ${E(\zeta_m) = L(\zeta_m)}$, so that ${LE/E}$ is cyclotomic

3. ${\mathfrak{p}}$ is unramified in ${LE/k}$

Moreover, we can choose ${m}$ such that it is divisible only by arbitrarily large primes.

The proof of this will use the previous number-theory lemmas and the basic tools of Galois theory.

So, first of all, we know that ${E}$ is a subextension of some ${L(\zeta_m)}$. We don’t know what ${m}$ is, but pretend we do, and will start carrying out the proof. As we do so, we will learn more and more about what ${m}$ has to be like, and eventually choose it.
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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.
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