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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