good reference for studying valuation theory (algebra)

Question

I need some good reference for studying valuations, I was looking for some really detailed material, as I'm just starting to study this theme. Please help.

Answer 1

Chapters 5,9 and 10 of Atiyah--Macdonald are a good algebraic reference http://www.math.toronto.edu/jcarlson/A--M.pdf

But as a number theorist I prefer Chapter II of Neukirch https://www.cimat.mx/~luis/seminarios/Teoria-de-Numeros/Neukirch_Algebraic_number_theory.pdf

Answer 2

Most books on Commutative Algebra should cover the basics, for example "Introduction to Commutative Algebra" by Atiyah and MacDonald, or "Commutative Ring theory" by Matsumura (not to be confused with his older book "Commutative Algebra").

Answer 3

It most definitely depends on what you are trying to do with valuations.

If you are trying to learn the basic ropes of algebraic number theory (class field theory and the like) then the references that were given to you by bounceback and red_trumpet are some of the best.

I might add the classic The Theory of Classical Valuations, by Paulo Ribenboim. It is very complete and has many extensive discussions on the many aspects of rank one valuations, including archimedean absolute values of number fields, ramification, Witt vectors, henselian fields etc.

If you are interested in higher rank valuations (also called Krull valuations sometimes, e.g. in the Ribenboim), then you need to gather several references:

1. Commutative Algebra, by Bourbaki, Chapter VI. It's quite technical, but it covers most of the basics.
2. Valuations and Local Uniformization, by Michel Vaquié. It's a fair introduction to valuation theory that makes use of Bourbaki at times, but is very complete. Its goal is to use valuation theory to illustrate the problem of local uniformization in some simple cases. The text itself may be a bit hard to swallow, but it is a concise introduction.

You might need to go into Commutative Algebra by Zariski and Samuel, but don't bother reading that presentation from the get-go as it's a pretty dated presentation.

Good luck.