Good references for Kronecker-Weber Theorem?

Question

I am looking for references to self-learn the proof of the Kronecker-Weber theorem in a way which takes the least prerequisites and can be understood easily and nicely. Please give your suggestions for references and explain why they are good. Also, it would be great if you could tell me what the central topics/ideas would be in any (or most) of these approaches.

Answer 1

I wrote my Bachelor's thesis on the Kronecker-Weber Theorem, so I've asked this question a lot of times. The standard way, as @DietrichBurde mentions, is to prove the local version, which is a lot easier because $p$-adic field extensions are very simple. For example, you never have problems of prime ideals splitting, as you only have the one, and whenever the prime ideal is unramified in an extension, your extension is immediately cyclotomic (!!).

The canonical reference to my knowledge is Washington's book Introduction to Cyclotomic Fields, in particular chapter 14, which is almost entirely self contained. A slightly more fleshed out version of the same proof is given by Sutherland in his Number Theory I course notes, freely available on MIT's website https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/pages/lecture-notes/. Lecture 20 is the proof. He gives more details, but also refers back to the previous lecture notes (lecture n for $n<20$). The final lemma for the proof he leaves as an exercise, which is a bit annoying - here Washington is better.

The notes of Dietrich Burde are also based on the proof of Washington.

If you have no experience with $p$-adic fields, this can seem a bit daunting, though. If you do not wish to study $p$-adics at the moment (though at some point it will most likely become impossible to avoid), there is a proof sketched in the exercises of Marcus' Number Fields chapter 4, completely avoiding the local case, using higher ramification theory. It requires a good bit of work to fill out the gaps, but he guides you pretty nicely through it. It is exercise 29 through 36.

Answer 2

See this exposition:

M. J. Greenberg (1974) An Elementary Proof of the Kronecker-Weber Theorem, The American Mathematical Monthly, 81:6, 601-607, DOI: 10.1080/00029890.1974.11993623