Some questions about Class Field Theory from a beginner

Question

I have started studying Class Field Theory. I have completed courses in algebraic number theory and commutative algebra. I have also done a reading project on $p$-adic numbers and the ramifcation of finite-degree extensions of $\mathbb{Q}_p$.

So far in my study, I have been able to complete the proof of the Global Kronecker-Weber Theorem (without using any local theory).

I have the following question:

Why is the Kronecker-Weber Theorem considered a precursor to Class Field Theory ?

I looked at the book Class Field Theory by Jurgen Neukirch. In the preface the author describes that there is a debate whether or not to use Cohomology in the study of Class Field Theory.

I have the following questions:

1. How is cohomology helpful in the study ?

2. Which parts of Class Field Theory are accessible with the help of cohomology ?

3. Is there any advantage of the cohomology based approach ?

Answer 1

Allow me to "add my two cents" to the detailed answer given by @ Adam Hughes. He rightly stresses that "if you ever wanted to do something at a full research level, certainly you should know as much as you can", and so you must get acquainted with both the two main approaches to CFT (I put apart Neukirch's, which I dare say is not useful to a beginner):

1) The approach via ideals, which was the first historically, starting with Kronecker-Weber (1886-87) and culminating with Takagi's main theorems (1920) on "K-modulii", "ray class fields", "conductors", ramification, decomposition ... , and Artin's reciprocity law (1927). In the excellent account by D. Garbannati, CFT summarized, Rocky Mountain J. of Math. 11, 2 (1981), this period is referred to as "classical global CFT". In this approach, local CFT is derived from global CFT. To get an idea of how to use this classical machinery, see e.g. G. Gras' book, CFT:From theory to practice, Springer (2005)

2) The subsequent developments via Chevalley's "idèles", referred to as "post - world war CFT" (op. cit.), in which global CFT is derived from local CFT. Not only Local CFT is easy (M. Hazewinkel, Adv. in Math. 18 (1975)) but the classical global CFT presents "aesthetically unpleasing aspects" (Garbanati, op. cit.), such as the "defining modulii" which vary with the finite abelian extensions of a given number field K and prevent to go smoothly to infinite extensions (such as in K-W's theorem).

Actually this change of perspective goes beyond mere technique, it ingrains in CFT (as more generally in arithmetic) the so called "local-global principle", which asserts, roughly speaking, that a certain property (not all !) over a number field K holds globally iff it holds locally over all its completions (p-adic as well as archimedean). This is where cohomology comes into play, because it appeared as the most convenient way to make the local-global machinery work. Essentially, the main new ingredient is the Brauer group of K. When expressed cohomologically, the local-global principle applies to it in a crystal clear manner.

One last word about cohomology, which nowadays completely pervades both algebraic and arithmetic geometry. Practically, you can consider it as a mere tool, at the same level as e.g. Taylor's formula in analysis, i.e. an approximation device. Taylor's expansion allows to approximate the value of an analytic function. The cohomological functor does the same for an algebraic object, by deriving from a short exact sequence an infinite exact sequence of cohomology groups

Answer 2

To your questions:

• Why is the Kronecker-Weber Theorem considered a precursor to Class Field Theory ?

  Class Field theory is concerned with reciprocity laws which are intimately related to the class group and Galois group of the maximal, unramified abelian extension of a given field. Naturally it is useful to understand what abelian extensions generally look like, especially for $\Bbb Q$, and Kronecker-Weber provides an invaluable basis for this, especially as we have some important theorems on Galois groups of composite extensions given information on the base Galois groups. Especially since subfields of cyclotomic ones are so simple to describe with some simple systems of congruences, knowing any given abelian extension of $\Bbb Q$ has a certain form can reduce your work-load significantly.

• How is cohomology helpful in the study ?

  This is a bit hard to quantify, since "helpful" is not a very precise word. Suffice to say cohomological methods are powerful and compact ways to express what would sometimes be much messier in other language, just like some theorems on modular arithmetic are much easier to state in the language of group theory. It expands your toolbox in a very fruitful way.

• Which parts of Class Field Theory are accessible with the help of cohomology ?

  Pretty much all of it, the book by Artin and Tate based off of lecture notes doesn't even include the first three chapters on cohomology and starts right with the cohomological approach to the theory.

• Is there any advantage of the cohomology based approach ?

  Yes, or we probably wouldn't use it. As before, there are a lot of well-studied, classical maps between important algebraic objects in number theory. Cohomology helps understand those maps, their images, kernels, and how they interplay with one another. Since those maps relate to things like the class group and Galois groups, cohomology is a natural approach to analyzing them in a systematic way with a lot of powerful tools and theory developed to further that approach.

Finally let me answer a question you don't ask:

• Should I focus my study of class field theory on the cohomological approach?

  That is a tough question and should be based on your comfort with it. If you ever wanted to do something at a full research level, certainly you should know as much as you can, but eg. Lang's Algebraic Number Theory treats things very well without direct mention of cohomology, at least for the basic theory in a first book. Neukirch takes a somewhat unorthodox approach using high level group theory to develop things, and Artin-Tate goes cohomological. I always recommend you start where your strengths are and explore as you go along to find the approach that most effectively enriches your understanding.