Prerequisites for algebraic number theory

Question

I am an undergrad who wants to study algebraic number theory. I saw that algebra is an essential part of algebraic number theory. So I studied algebra from Dummit & Foote and have recently completed commutative algebra from Atiyah Macdonald. I was thinking about starting to study algebraic number theory from Neukirch. It will be great if anyone can tell me, do I need to study some thing else to cover the prerequisites for Neukirch?

Answer 1

You have all the necessary prerequisites for Neukirch. However, it‘s still a very tough book, covering a basic course in algebraic number theory in about 60 pages, if I remember correctly, and going straight into class field theory after that—using an unusual approach, too.

A more detailed first text going at a more leisurely pace would be Milne‘s notes or Lang‘s book.

Answer 2

I would not recommend Neukirch; it’s tough and the main goal is Class Field Theory. The courses in Algebraic Number Theory I took at Berkeley barely gave the statements of the theorems of Class Field Theory at the end of the first semester, and it took most of the second to cover them.

I would strongly recommend Marcus’s Number Fields, from Universitext. It’s a very good book, with lots of good problems and exercises, and will cover the important topics (including a proof of FLT in the regular case as a series of exercises). It does not include Class Field Theory, but it will put you in a good position to jump into Class Field Theory when you are done.

Note also that Neukirch’s approach to Class Field Theory is a bit different from the most typical ones; in a sense, it “goes the other way” in establishing the correspondences.

Answer 3

For self study, the new book Quadratic Number Theory by J. L. Lehman is a good idea.

He does (binary) quadratic forms and quadratic number fields together. From my experience on MSE, this is a very good idea. Computations are faster in forms. By restricting to the quadratic case, he gets to a fair amount of algebraic number theory.

Answer 4

I second the suggestion by Arturo to read Number Fields by Marcus and I also suggest reading Algebraic Theory of Numbers by Samuel. Be sure to solve lots of the exercises in these books.