# Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped through the first pages and realized that I am not quite ready to read it. In particular, the two main prerequisites that I'm lacking seem to be:

Commutative algebra. I have read more than half of Atiyah-Macdonald. But I am having a hard time, because it's concise and somewhat unmotivated. Is there another book that is easier to read? Which topics are relevant to algebraic number theory?

Galois theory. I am practically blind about Galois theory. Same questions as before: what are good books in Galois theory, and which topics are relevant to algebraic number theory?

• The second half of the text by Dummit and Foote ("Abstract Algebra" I believe) is a good introductory text to a small bit of Galois theory. – zzzzzzzzzzz May 10 '13 at 15:47
• About Commutative Algebra, you might try to take a look at the notes you can find here jmilne.org/math/xnotes/ca.html – Marco Vergura May 10 '13 at 16:00
• For Galois theory, I recomment Aluffi's "Algebra. Chapter 0" – xyzzyz May 10 '13 at 16:01
• I found Janusz's "Algebraic Number Fields" very nice, but I remember it required quite a good general algebra (linear, abstract) background, though not necessarily commutative algebra. Eisenbud's "Commutative Algebra" is great but I'm not sure it's easier than A-M's. Try Reid's books: "Undergraduate Commutative Algbera", "Undergraduate Algerbaic Geometry" and, perhaps more important to you, "Galois Theory". Milne's books are great and free to download from the web, e.g.:jmilne.org/math/CourseNotes/ft.html – DonAntonio May 10 '13 at 16:03
• Try also Algebraic Theory of Numbers by Pierre Samuel. – lhf May 10 '13 at 17:04

Take a look at Keith Conrad's expository articles here http://www.math.uconn.edu/~kconrad/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://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 by Kleiman and Altman available here: http://stuff.mit.edu/afs/athena/course/18/18.705/www/syl12f.html, 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'd suggest not to "prepare" but just to dig in. Keeping in mind that if the going gets rough, you might need to look for outside help (catch up on some stuff you aren't familiar with, fill in some gaps). Either through other texts, looking for lecture notes (the 'net is chock full of them, some awful, a few outstanding), look for formal classes at e.g. coursera, or by bothering folks here.

"Preparing for" isn't very rewarding in itself. And you could easily "overlearn" stuff you won't need, or "misslearn" by not seeing how and where you will need it.

(Just my style. YMMV.)

Good luck!

• I second this. Especially because the generality of Galois theory is not especially high when first learning ANT (e.g. everything is presumably over $\mathbf{Q}$ to begin). – tkr May 10 '13 at 18:54

There are two books that are I think are good at helping with Galois Theory. One was already recommended by rckrd in the comments, which is Dummit & Foote's Abstract Algebra. It's a reasonable book with lots of exercises to work through. The other (after you've finished with Dummit & Foote) that I like is Galois Theory by Cox, although a number of criticisms I've heard about him is how much time he spends on polynomials. Your mileage may vary and how much that is an issue for you is probably more on personal preferences.

Another book that I picked up short while ago was Visual Group Theory by Nathan Carter. This book is not really about Galois Theory but has excellent visualization strategies for groups, which might help a bit.

No matter what sources you start with though, I think you will need to spend some quality time going through exercises and proofs to get a reasonable intuition for Galois Theory. Good luck!

If you are not really comfortable with Commutative algebra and Galois Theory and want to learn Algebraic Number Theory, I have two suggestions.

1. Introductory Algebraic Number Theory by Saban Alaca and Kenneth A Williams

2. Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart and David Tall.

These two books are very basic.