Hi readers,

I have been studying Class Field theory this semester. Since I didn’t take the Algebraic number theory course offered in the previous semester, I am facing difficulty following the course. But this in no sense makes the course less interesting. Here in this post, I pen down a few thoughts on the subject, which are just my understanding of the subject. Note that WHAT FOLLOWS MAY BE INCORRECT, not to mention imprecise. But I will try and update this as my understanding increases over time.

In Class Field theory (CFT), the mission is to understand \overline{\mathbb Q} / \mathbb Q which is horrendously difficult, so they study \mathbb Q^{\text{ab}} / \mathbb Q, where \mathbb Q^{\text{ab}} is the maximal abelian extension of \mathbb Q. By that, I mean such an (infinite) extension of \mathbb Q so that its Galois group (which again is infinite) is abelian. Now for infinite Galois extensions the correspondence between subfields and subgroups is no longer true, so they do what they always do — change the definitions to make this correspondence work. One defines a topology on \mathbb Q^{\text{ab}} and now instead of subsets of this huge (uncountable) set, the correspondence is between closed sets and subgroups.

One also studies the field extension L/K where L and K are number fields instead of K/Q, as in a basic number theory course. Initially, I had difficulty assimilating the results and finding which results true for K/Q go through to L/K. For example, it every extension K/Q, at least one prime ramifies (in fact, all the primes dividing the discriminant). The units of Z are just 1 and -1 but there are lots more units in \mathcal O_K (given by Dirichlet in his Units theorem). As a consequence, it is possible that in an extension L/K of number fields, no prime \mathfrak p of \mathcal O_K ramifies in \mathcal O_L. When the discriminant is a unit then no prime ramifies and the extension is called “unramified”.

To be continued..