Class Field Theory 2


Today we’ll start by sketching an example of a Hilbert class field. Let {L} be the splitting field of {x^3-x-1} over {\mathbb{Q}}. Thus {L=\mathbb{Q}(\alpha_1, \alpha_2, \alpha_3)} where {\alpha_j} are the roots of the polynomial. Standard Galois theory shows us that {Gal(L/\mathbb{Q})\simeq S_3}. I know, we assumed abelian extensions last time, but we aren’t interested in this extension. It turns out that one can also argue that we have a subextension {K=\mathbb{Q}(\sqrt{-23})\subset L}.

To see how useful this class field theory machinery is, we can check with some basic algebraic number theory (involves the Minkowski bound) that {Cl(\mathcal{O}_K)\simeq \mathbb{Z}/3}. But above we said that {Gal(L/K)\simeq \mathbb{Z}/3}. Thus by the main theorem last time we see that {L} must be the Hilbert class field of {K}, and hence {L} is the maximal unramified abelian extension of {\mathbb{Q}(\sqrt{-23})}. That’s pretty interesting that it is so small.

This gives us an idea of what these Hilbert class fields look like. Note also that class field theory tells us that if the class number is {1}, then there are no unramified abelian extensions, and hence fields like {\mathbb{Q}(i)} or {\mathbb{Q}(\sqrt{-11})} have Hilbert class field equal to themselves. The word “maximal” makes these things sound big, but being unramified is pretty strict and we see that in order to have large Hilbert class fields the class group needs to be large.

Let’s try to go the other direction now. Recall a few posts ago we calculated that {Cl_m(\mathbb{Q})\simeq (\mathbb{Z}/m)^*} when we include the embedding in the conductor. We have an isomorphism {Cl_m(\mathbb{Q})\simeq (\mathbb{Z}/m)^*\stackrel{\sim}{\rightarrow} Gal(\mathbb{Q}(\zeta_m)/\mathbb{Q})}, where {\zeta_m} is a primitive {m}-th root of unity. We’ll suggestively call the isomorphism {\Psi([a])=\sigma_a} where {\sigma_a(\zeta_m)=\zeta_m^a}. This comes from standard theory of cyclotomic fields.

We have that the ring of integers is {\mathbb{Z}[\zeta_m]}. But now on the primes, we see that {\Psi(p\mathbb{Z})} is just the Frobenius map of raising to the {p}. This shows us that our standard isomorphism is actually the Artin map {\Psi_{\mathbb{Q}(\zeta_m)/\mathbb{Q}}} and hence class field theory tells us that the ray class field of {\mathbb{Q}} of conductor {\frak{m}=\{(m), \mathbb{Q}\hookrightarrow \mathbb{R}\}} is just the cyclotomic extension {\mathbb{Q}(\zeta_m)}.

This is an incredibly important example, because now let’s apply the Galois correspondence. Let {L/\mathbb{Q}} be any abelian extension at all. We know that {L} corresponds to some conductor {\frak{m}_L}. This must be of the form {\{\mathbb{Z}m\}} or {\{\mathbb{Z}m, w_1\}}. Since the first one divides the second, we get two surjections {Cl_{\{(m), w_1\}}(\mathbb{Q})\twoheadrightarrow Cl_\frak{m}(\mathbb{Q})\twoheadrightarrow Gal(L/\mathbb{Q})}. But the first term of this by the above is that {Cl_{\{(m), w_1\}}(\mathbb{Q})\simeq Gal(\mathbb{Q}(\zeta_m)/\mathbb{Q})}. Therefore, {L\subset \mathbb{Q}(\zeta_m)}. Out of class field theory we get the classical result called the Kronecker-Weber theorem that any abelian extension of {\mathbb{Q}} must be contained inside a cyclotomic field.

To finish today let’s talk about one more topic. Suppose we have our abelian extension {L/K} of number fields with Galois group {G}. If we have a further subextension {K/F}, then we get {Gal(L/F)=H} is possibly non-abelian. But we can take the abelianization {H^{ab}=H/[H,H]} and this corresponds via Galois theory to a maximal abelian subextension say {L'} between {L} and {F}, i.e. {Gal(L'/F)\simeq H^{ab}}. By the universal property of the abelianization, we have a map {G\rightarrow H^{ab}} since {G\subset H}.

The point is that {G} corresponds to an abelian extension {L/K} and so class field theory over {K} tells us something about it, and {H^{ab}} corresponds to an abelian extension {L'/F}, so class field theory over {F} tells us something about it. We just produced a natural map {G\rightarrow H^{ab}}, and hence there should be a corresponding statement in class field theory.

Here’s the theorem. Let {\frak{m}=\frak{m}_{K/F}} so that {\frak{m}_L=\frak{m}_{L/K}} divides {\frak{m}} and {L/F} is unramified outside the places of {F} under {\frak{m}}. Then we get a commutative diagram:

\displaystyle \begin{matrix} Cl_\frak{m}(K) & \twoheadrightarrow & Cl_{\frak{m}_L}(K) & \stackrel{\Psi_{L/K}}{\rightarrow} & G & \rightarrow & 1 \\ id \downarrow & & & & \downarrow & & \\ Cl_\frak{m}(K) & \stackrel{N_{K/F}}{\rightarrow} & Cl_{\frak{m}_{L'}}(F) & \stackrel{\Psi_{L'/F}}{\rightarrow} & H^{ab} & \rightarrow & 1 \end{matrix}

We could call this a functoriality property of the Artin map. The loose description of this is that inclusions of Galois groups go to the corresponding norm map. We’ll pick up here next time.

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