Some Basics of Class Field Theory

Class field theory is one of the crown jewels of modern algebraic number theory. It can be viewed as a generalization of the “reciprocity laws” discovered by Carl Friedrich Gauss and other number theorists of the 19th century. However, as we have not yet discussed reciprocity laws in this blog, we will leave that point of view to the references for now. Instead, in order to describe class field theory, we rely on the following quote from the mathematician Claude Chevalley:

“The object of class field theory is to show how the abelian extensions of an algebraic number field $K$ can be determined by objects drawn from our knowledge of $K$ itself; or, if one prefers to present things in dialectic terms, how a field contains within itself the elements of its own transcending.”

Our approach in this post, as with much of the modern literature, will start from the “local” point of view, and then we will put together all these local pieces in order to have a “global” theory. Recall that there is an analogy between geometry and number theory, and that primes play the role of points in this analogy. Therefore, as in geometry “local” means “zooming in” on a point, in number theory “local” means “zooming in” on a prime. Putting these pieces together is akin to what we have done in Adeles and Ideles. This will become more clear when we define local fields and global fields.

Let $K$ be a (nonarchimedean) local field. This means that $K$ is complete with respect to a discrete valuation (see Valuations and Completions and Adeles and Ideles) and that its residue field is finite. These are either the fields $\mathbb{Q}_{p}$ (the $p$-adic numbers), $\mathbb{F}_{p}((t))$ (the field of formal power series over a finite field $\mathbb{F}_{p}$), or their finite extensions. Let $L$ be a finite extension of $K$.

We define the norm homomorphism as

$\displaystyle N_{L|K}(x)=\prod_{\sigma}\sigma x$

for $x\in L$ and $\sigma\in \text{Gal}(L|K)$ (note that there are many notations for the action of $\sigma$ on $x$; in the book Algebraic Number Theory by Jurgen Neukirch, the notation $x^{\sigma}$ is used instead). We let $N_{L|K}L^{\times}$ stand for the image of the norm homomorphism in $K$. Then local class field theory tells us that we have the following isomorphism:

$\displaystyle K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}$.

We see that everything in the left-hand side belongs to the field $K$. This is related to Chevalley’s quote earlier. However, there is even more to this, as we shall see later.

Understanding more about this isomorphism requires the theory of Galois cohomology (see Etale Cohomology of Fields and Galois Cohomology). Namely, the Galois cohomology group $H^{2}(\text{Gal}(L|K),L^{\times})$ is isomorphic as a group to the group homomorphisms from $\text{Gal}(L|K)^{\text{ab}}$ to $K^{\times}/N_{L|K}L^{\times}$. It is cyclic of degree equal to the degree of $L$ over $K$.

There is an injective map from $H^{2}(\text{Gal}(L|K),L^{\times})$ to the quotient $\mathbb{Q}/\mathbb{Z}$, and the element of $H^{2}(\text{Gal}(L|K),L^{\times})$ that gets mapped to $1/n$, where $n$ is the degree of $L$ over $K$, is precisely the element that corresponds to the inverse of the isomorphism $K^{\times}/N_{L|K}L^{\times}\xrightarrow{\sim}\text{Gal}(L|K)^{\text{ab}}$.

Now let $K$ be a global field, which means that it is a finite extension either of $\mathbb{Q}$ (the rational numbers) or of $\mathbb{F}_{p}(t)$ (the function field over a finite field $\mathbb{F}_{p}$). Let $L$ be a finite extension of $K$. Let $C_{K}$ and $C_{L}$ denote the idele class groups (see Adeles and Ideles) of $K$ and $L$ respectively. As in the local case, we will need a norm homomorphism, but this time it will be for idele class groups.

We will define this norm homomorphism “componentwise”. Writing an idele as $(z_{w})$, we take the norm $N_{L_{w}|K_{v}}(z_{w})$, and take the product for all primes $w$ above $v$. We do this for every prime $v$ of $K$, and thus we obtain an element of the group of ideles of $K$, and then we take the quotient to obtain an element of the idele class group of $K$. We denote by $N_{L|K}C_{L}$ the image of this norm homomorphism in $C_{K}$.

Then global class field theory tells us that we have the following isomorphism:

$\displaystyle C_{K}/N_{L|K}C_{L}\sim\text{Gal}(L|K)^{\text{ab}}$

Again, as in the local case, everything in the left-hand side belongs to $C_{K}$.

As we have said earlier, we can obtain this isomorphism by putting together the local pieces from local class field theory, i.e. homomorphisms from $K_{v}^{\times}$ to $\text{Gal}(L_{w}|K_{v})^{\text{ab}}$ which come from the isomorphisms from $K_{v}^{\times}/N_{L|K}L_{w}^{\times}$, as ideles have components which are local fields, and then taking the quotient to obtain the isomorphism for idele class groups, similar to what we have done for the norm homomorphism.

However, in order to obtain the desired isomorphism, the map (called the Artin map)

$\psi:I_{K}^{\times}\rightarrow\text{Gal}(L|K)^{\text{ab}}$

from the group of ideles $I_{K}$ of $K$ to the group $\text{Gal}(L|K)^{\text{ab}}$, which is obtained from putting together the local pieces (before taking the quotients) must satisfy three properties:

(i) It has to be continuous with respect to the topologies on $I_{K}$ and $\text{Gal}(L|K)$ (the topology on the group of ideles is discussed in Adeles and Ideles, while the topology on $\text{Gal}(L|K)$ is the so-called Krull topology – the latter is part of the theory of profinite groups).

(ii) The image of $K^{\times}$ (as embedded in its group of ideles $I_{K}$) is equal to the identity.

(iii) It is equal to the Frobenius morphism for elements in $I_{K}^{S}$ (see again Adeles and Ideles for the explanation of this notation), where $S$ consists of the archimedean primes and those primes which are ramified in $L$ (see Splitting of Primes in Extensions).

It is a challenging task in itself to prove that the Artin map does indeed satisfy these properties, and for now we leave it to the references. Instead, we mention a few more properties of class field theory. In the local case, class field theory also classifies the subgroups of $K^{\times}$ which are of the form $N_{L|K}L^{\times}$, which correspond to the open subgroups of finite index in $K^{\times}$. Since the finite abelian extension $L$ of $K$ also obviously corresponds to the subgroup $N_{L|K}L^{\times}$, we then obtain a classification of the finite abelian extensions of $K$. Similarly, in the global case, class field theory classifies the subgroups of $C_{K}$ which are of the form $N_{L|K}C_{L}$, which correspond to the open subgroups of finite index in $C_{K}$. The field which corresponds to the such a subgroup is called its class field. In the case that $L$ is the maximal unramified abelian extension of $K$, $L$ is called the Hilbert class field of $K$, and there we have the result that the ideal class group (see Algebraic Numbers) of $K$ is isomorphic to the Galois group $\text{Gal}(L|K)$. With the ideas discussed in this last paragraph, the goal of class field theory as expressed in the quote of Chevalley is fulfilled; we are able to describe the abelian extensions of $K$ from knowledge only of $K$ itself.

References:

Class Field Theory on Wikipedia

Artin Reciprocity Law on Wikipedia

Profinite Group on Wikipedia

Class Field Theory by J.S. Milne

Algebraic Number Theory by Jurgen Neukirch

Algebraic Number Theory by J. W. S. Cassels and A. Frohlich

A Panorama of Pure Mathematics by Jean Dieudonne

Primes of the Form $x^{2}+ny^{2}$ by David A. Cox