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.

According to local class field theory, this isomorphism holds for {L/k} abelian more gneerally, but it is harder to describe.

Now suppose {L} and {k} are number fields. The group {k^*/NL^*} will generally be infinite, so there is no hope for such an isomorphism as before. The remedy is to use the ideles. The fundamental theorem of class field theory is a bijection between the open subgroups of {J_k/k^*} of finite index and the abelian extensions of {k}. For a finite abelian extension {L/k}, class field theory gives an isomorphism

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

What exactly is the norm on ideles though? I never defined this, so I may as well now.

Note that the Galois group {G(L/k)} maps each completion {L_w} to {L_{\sigma w}} in some manner (when {\sigma} is in the decomposition group, then this is just the usual action on {L_w}). In this way, {G(L/k)} acts on the idele group, and even on the idele classes because the action reduces to the usual action on {L^*}. So the norm is defined by {N(i) = \prod_{\sigma \in G(L/k)} \sigma i}.

This result, called the Artin reciprocity law, is somewhat complicated to state and takes a large amount of effort to prove. In outline, here will be the strategy.

First, we will compute the order {J_k/k^* NJ_L }. We will prove that it is equal to {[L:k]} by proving the {\leq} and {\geq } inequalities. The {\geq} proof uses some cohomological machinery (the Herbrand quotient) and a detailed look at the cohomology of local fields and units. Specifically, one shows that the Herbrand quotient (which divides the order of {J_k/k^* NJ_L }) is equal to {[L:k]} (at least for a cyclic extension). The {\leq} proof uses a nifty trick manipulation in the land of analysis, specifically the basic properties of L-functions.

We then describe a map, called the Artin symbol, from the ideles to the Galois group and prove it surjective. The Artin symbol sends an idele which is a uniformizer at an unramified valuation and sufficiently 1 everywhere else to a Frobenius element with respect to said valuation, though more generally it is harder to describe. By the dimension count, we have proved the isomorphism. We will then show that any finite-index subgroup corresponds to some field by reducing to a special case (where enough roots of unity are contained) and then give explicit constructions of abelian extensions using Kummer theory.

This is a very loose outline of the proof that utterly ignores many details and subtleties, but these will be discussed in due time.  It follows the approach in Lang’s Algebraic Number Theory, but I think I’ll probably try to avoid the “generalized ideal class groups” and stick to ideles.

There is an entirely different, and many say better, approach via cohomology. The idea is to define the local Artin map using a bit of cohomological magic called “Tate’s theorem” that drops out the isomorphism of local class field theory.  Then, you piece together all these local maps k_v/NL_w \to G_w into a global map on the completion. The hard part is to show that it factors through k^*.

This approach still needs the proof of the first and second inequalities, but develops local class field theory before global class field theory.  I’m not going to follow it because I could never really understand what was going on in Tate’s theorem.  Also, I’m not that comfortable with cup-products in group cohomology, and sadly there don’t seem to be many good sources, thoughI’ll try looking at Weiss’s Cohomology of Groups.

Anyone who reads this should definitely see this MO question and the answers, many of which are from actual mathematicians.

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