Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't need that. I'm seaching for book which motivates commutative algebra using algebraic number theory.My main is to learn algebraic number theory but while doing so I also want to pick up enough commutative algebra to deal with algebraic geometry as well.
[Math] Book recommendations for commutative algebra and algebraic number theory
algebraic-geometryalgebraic-number-theorycommutative-algebrareference-request
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Take a look at Keith Conrad's expository articles here https://kconrad.math.uconn.edu/blurbs/ on Galois Theory and algebraic number theory. They are all wonderful. In particular, he shows you how to compute stuff using the tools you learn which is very essential in my opinion. Most of the topics on Galois Theory covered in his notes seem relevant to algebraic number theory, but I haven't read all of them in detail because I learned Galois theory in a college course. There is a fair bit of elementary algebraic number theory (at the level of Samuels's book mentioned by lhf above) which you will need to be fairly comfortable with before you move on to to class field theory. I think Keith's notes on algebraic number theory covers this elementary material well, with many examples of computations. Finally, I highly recommend his note "The History of Class Field Theory". I am certain you will be adequately prepared to read Lang if you work through some of Conrad's material.
For algebraic number theory, I also recommend Cassels-Fröhlich's "Algebraic Number Theory" and Cox's "Primes of the form $x^2 + ny^2$". James Milne's notes on algebraic number theory and class field theory, freely available on his website, are great. The other standard reference is Neukirch's "Algebraic Number Theory" which I personally really like. When you read about valuations, completions etc., I recommend the handouts from Pete L. Clark's course available here: http://alpha.math.uga.edu/~pete/MATH8410.html. Also, highly recommended are Serre's "Local Fields" and Iwasawa's "Local Class Field Theory" (the latter is harder to find, but I have a pdf copy which I am willing to share with you). You see from this that there are many good choices, and you will have to choose your own poison.
I should mention that there are different approaches to class field theory (discussed in the following link: https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first), and I learned local class field theory through Lubin-Tate formal groups. Their short paper on this topic is a wonderful read. Luckily, Prof. Lubin frequents MSE so he may be able to recommend more sources.
I think that if you want geometric motivation for commutative algebra, you will need some knowledge of algebraic geometry (at least at the level of classical varieties) and DonAntonio's suggestions are great. However, you can learn the required commutative algebra as and when you need it. Certainly books like Cassels-Fröhlich prove most of the results of commutative algebra used in algebraic number theory along the way.
If you want to learn commutative algebra as a subject in its own right, take a look at Eisenbud's "Commutative Algebra with a View Toward Algebraic Geometry" which motivates the algebraic constructions using geometry. However, as mentioned in the previous paragraph, you may need some basic knowledge of classical varieties to understand the geometric motivations in this book. For this, I recommend Karen Smith's "An Invitation to Algebraic Geometry". Also, there is this new gem A term of Commutative Algebra by Altman and Kleiman, which is like an Atiyah-Macdonald 2.0. In particular, the authors mention that their aim is to improve some of the exposition in Atiyah-Macdonald using categorical language. The notes are really wonderful.
Despite these recommendations, I don't think you need so much preparation to read Lang. In particular, I came across the various texts mentioned in my answer as and when I needed to learn a particular topic.
I am telling my favourites-
- Introduction to Set Theory- Karel Hrbacek, Thomas Jech
- Real analysis- Howie and Understanding analysis- Abbott Complex analysis- Zill & Shanahan and Complex analysis - Lang
Abstract algebra- Dummit & Foote and Contemporary Abstract algebra- Gallian
Linear Algebra- Friedberg, Insel and Spence & Linear Algebra- Hoffman & Kunze
- Differential equations- Sheldon Ross No idea about PDE, but you can check out Simmons book
- Topology- Munkres or Topology- Dugundji. Dont know about algebraic topology but see
- An Introduction to Manifolds- Loring Tu
- Undergraduate Commutative Algebra- Miles reid (for beginner/self study) Commutative algebra - Atiyah (Classic) Steps in Commutative Algebra-R.Y.Sharp. (I like it)
Hartshorne's Algebraic Geometry
A Classical Introduction to Modern Number Theory- Ireland & Rosen
- Introduction to Lie Algebras-Erdmann & Wildon
Best Answer
I agree with David Loeffler's answer: there is a large initial segment of algebraic number theory which essentially coincides with the study of Dedekind domains. A careful study of Dedekind domains gives an introduction to several important commutative algebra topics: e.g. localization, integral closure, discrete valuations, fractional ideals, and the ideal class group.
So one can motivate much of basic commutative algebra using concepts from algebraic number theory, but there is also a lot missing, for instance:
$\bullet$ Module theory. Modules over a Dedekind domain are "too nice" compared to modules over an arbitrary commutative ring. For instance injective = divisible and flat = torsionfree.
$\bullet$ The spectrum. The family of prime ideals in a Dedekind domain is unrepresentatively simple: all the nonzero ones are maximal. This is not a good motivation for spending time understanding the order-theoretic structure or the Zariski topology on $\operatorname{Spec} R$.
$\bullet$ Dimension theory.
$\bullet$ Primary decomposition. One can view primary decomposition in a Noetherian ring as a generalization of factorization of ideals into products of primes in a Dedekind domain, but once again the former is significantly more complicated than the latter.
$\bullet$ The Nullstellensatz.
Rather, if you study algebraic number theory and algebraic geometry at more or less the same time, you'll see that much of what you're doing is commutative algebra and that algebra will be well motivated. Among reasonably introductory texts I know of exactly one that pulls this off well: this text by my colleague Dino Lorenzini.
(Since my own commutative algebra notes have been mentioned, let me say that I view these notes as being at approximately the level of a student who has had a first, relatively nontechnical, course in either algebraic number theory -- e.g. from Marcus's text -- or algebraic geometry -- e.g. from Shafarevich's text -- and has been told that she needs to learn some commutative algebra before proceeding onward. On the other hand, my notes draw more explicitly on examples from topology and geometry than from either of the aforementioned areas.)