## Part 1 – The main ideas of Global Class Field Theory

I really wanted to get started on something meaty so apologies to people following my posts on the basics of algebraic number theory…I will try and sort it out sometime in the future.

This is actually going to be a short post compared to the usual. I wanted to start my exposition of class field theory by giving a brief glimpse of its ideas and its power.

As we have discussed, we like number fields in algebraic number theory. We usually classify them by degree. Now say we start with a specific number field $K$…how might we classify the extensions of $K$?

Well we have the crude algebraic/transcendental classification but this really isn’t telling us much. Let’s be more specific.

How might we classify the algebraic extensions? Well we can do it by degree but again, extensions of the same degree can behave very differently (for example some have abelian Galois group, some non-abelian…some aren’t even Galois extensions!).

Let’s specialize again. Ok so how might we classify Galois extensions of $K$? This might be starting to sound silly, because really we only know about the degree as a measure of size.

What is completely counter-intuitive is that if we make one more step and go to abelian extensions (ones with abelian Galois group), there is a way to classify all such extensions of $K$…in terms of the arithmetic of $K$ itself.

This is strange, how could it possibly be that the field you start with somehow determines how you may extend? What is the classification based in terms of?

Well these are the questions that global class field theory answers. It describes certain extrinsic notions attatched to $K$, i.e. the abelian extensions of $K$, in terms of certain intrinsic notions attatched to $K$, i.e. certain subgroups of fractional ideals of $K$. We will build this correspondence more precisely and see it in action.

Really, it turns out that we are classifying abelian extensions in terms of the ramification in a nice way. Frobenius elements will play a huge role here.

Once we have seen the theorems we will see that we will be able to tell a lot about a given extension from the correspondence…for example which primes split completely in the extension, something about how the primes are distributed, etc. We will also be able to in some sense construct certain “maximal” extensions of a number field with given ramification. These will answer nice questions in number theory, for example in answering the question “Which primes can be written in the form $x^2 + ny^2$ for positive square-free integer $n$?”.

On the other hand we will see that one of the main theorems of class field theory, the Artin reciprocity law, really is the most general reciprocity law we know, bringing all other reciprocity laws into one. This theorem sets in stone one direction of the correspondence mentioned above.

We will also see the Takagi existence theorem, which gives the other direction of the correspondence, and the Cebotarev density theorem, which is a far reaching generalisation of Dirichlet’s theorem on primes in arithmetic progressions. If we are lucky we will show how Dirichlet’s theorem is a simple corollary of this theorem.

We will mainly stick to the theory in terms of ideals but I may choose to say something about the theory of adeles and ideles, certain constructions where we view all completions of a number field at once. The classification is apparently much smoother in terms of ideles but I have never got on with this (too much topology for my liking).

There are many good books for approaching this theory, namely Cox “Primes of the form $x^2 + ny^2$” for a survey of the area, Childress “Class field theory” for a fully proved account of class field theory and Lang “Algebraic number theory” for those who like Lang. There are also my own papers on this, they can be found on my academia page.

I should maybe say something about the current state of affairs. I am not at this stage in my development yet but I believe that the Langlands program (usually referred to as “non-abelian class field theory”) is an attempt to extend the notions in class field theory to general Galois extensions (so that we get an even better correspondence of more things).

This has some kind of connection to Galois representations (turns out that class field theory studies one dimensional Galois representations into “$\text{GL}_1$” of the adeles (which are the ideles). This is abelian, which ties in with the classification of abelian extensions.

There are far reaching generalisations of the theory (which I am led to believe form the basis of the Langlands program mentioned above)  of the higher dimensional  Galois representations into “$\text{GL}_n$” of the adeles! These things are non-abelian and so it is believed that they will classify non-abelian extensions but at this point in time this has not be done in general.

It is said that it takes most people at least a year to understand class field theory properly, and I can certainly vouch for this…but I hope to at least give some indication of the main ideas. Hopefully you will all enjoy the journey and maybe discuss things with me so that I may learn new stuff too!