# References for Algebraic Number Theory and Class Field Theory

My long term goal for this "reading/study project" is to understand roughly what the Langlands conjectures are about. A more modest short term goal though, and more realistic one, is to understand:

1) the basics of Algebraic Number Theory, with a lot of examples worked out,

2) the basics of class field theory, but explained from a modern point of view, using adeles and ideles, eventually. However, I would like the abstraction to be gradual, so to speak, and motivated by a few worked out examples.

My background includes of course the usual Graduate Algebra courses, but I did read on my own quite a bit of commutative algebra and algebraic geometry. So I know what is a Noetherian ring, localization, the ring of integers in a number field, a UFD, a PID, the Galois group etc. I would like to know more about Dedekind domains and "onwards".

I would possibly like references that make analogies with algebraic geometry say, via schemes, for instance (I know the basics of schemes but I prefer an approach with many examples, kind of like, say, Eisenbud and Harris's book "The Geometry of Schemes", but with more examples worked out on the Algebraic Number Theory side).

I realize that my requirements may not all be met at once, but I will take whatever I can get, so to speak, in terms of advice and recommendations. Thank you!

• From what you're describing, Neukirch's book could be a good start. Or Milne's notes – Mathmo123 May 16 '18 at 15:54
• – lhf May 16 '18 at 15:54
• J.S. Milne's Algebraic Number Theory and Fields and Galois Theory – emma May 16 '18 at 16:00
• And, when you get to it, Benedict Gross's (youtube) Eilenberg lectures are a great intro to the Langlands program.... – peter a g May 16 '18 at 16:02
• @Mathmo123, yes, I have stumbled on Milne's notes, and love them. Thank you for Neukirch's book. I will check it out. lhf, thank you, yes I just checked 2 posts which ask for references for class field theory. It seems each one would like to learn things in a certain way, but only a limited number of books/approaches are available. – Malkoun May 16 '18 at 16:03