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I would appreciate any suggestions for book or notes on ANT at a level that I would characterize as advanced beginner. I.e., something assuming familiarity with topics in Dummit & Foote, that is a little less than Samuel or Marcus.

I would especially like accessible discussions of topics such as fractional ideals, ramification, and ideal classes.

Any suggestions would be appreciated. (I have seen several enthusiastic endorsements of Stewart and Tall.)


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Serre, Local fields. – Adeel Apr 3 '13 at 15:00
up vote 6 down vote accepted

I really think Algebraic Number Theory by Nuekirch is incredible. Very thorough and not too hard to pick up. Problems in Algebraic Number Theory by Murty and Esmonde is another great book to work through problems. Also, William Stein has a great set of notes online that give a computational approach to the subject. I'd also like recommend A Course in Computational Algebraic Number Theory as a companion guide. Not much theory, but after you get the basics down it's great for computing

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You're right about Neukirch; the few sections I could understand - Gaussian integers and p-adic numbers, are exceptional. Unfortunately, I am not ready for other material. – TheBirdistheWord Apr 3 '13 at 16:03
Just bought a copy of Murty and Esmonde (upon a friend's recommendation). So far I have been enjoying it, but no final nod, yet. – Jyrki Lahtonen Apr 5 '13 at 21:14
@JyrkiLahtonen Hi, Jyrki - I would be interested in your ultimate opinion. I hadn't seen you comment, so if you don't mind, maybe you would send any further remarks to me too. Thanks very much. Andrew – TheBirdistheWord Apr 15 '13 at 20:17

Ted Chinburg has videos of his lectures for what is going on a 2 year course in algebraic number theory online.

Direct link: Semester 1, Semester 2 , Semester 3 and Semester 4

Also, there's the MSRI database for all the things that go on there, they're all over the website at each program's site.

Source: Mathoverflow.

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These look great. Lots over my head for now, but will be fabulous later on.Thanks. – TheBirdistheWord Apr 3 '13 at 15:04

Some online notes:

Robert B. Ash

J. Milne

Neukirch and Stein have already been mentioned, but I'd second both of them, Stewart and Tall is good too!

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Seconding the suggestion of Milne's notes. – Potato Apr 3 '13 at 17:59
+1 for thanks; but I would put Milne (good as he is) at the same level of difficulty as Samuel and Marcus. – TheBirdistheWord Apr 3 '13 at 21:59

I will provide, for anyone who subsequently comes across this question, what I am finding most beneficial.

These are pieces most generously posted by Keith Conrad:

What makes them especially remarkable for me as a self-studier is the clarity of presentation, the richness of examples, and the frequent warnings that point out possible misconceptions. Plus the exposition is quite motivating and keeps you engrossed.

They cover a wide range of topics (I am focusing on the Algebraic Number Theory section).


I just came across a great set of notes by Franz Lemmermeyer:

They a beautifully written and motivating, including numerous comments on the historical background involved in the development of the material.


I appreciate the helpful advice offered, and have looked into them. However, I feel the items I've posted are of great value to me, so I wanted to emphasize that sentiment.

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I self study ANT myself and use and I looked at many other books, but for a beginner in ANT, I find these very accessible. Prerequisites are only Elementary Number Theory and Linear Algebra.

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Hi - You may also be interested in these excellent, complete notes from a course given at Univ. of Chicago elaborating on the main ANT chapters in "Ireland & Rosen." – TheBirdistheWord Oct 7 '14 at 22:01

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