# Beginner's text for Algebraic Number Theory

What's good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done an basic abstract algebra course.

• Marcus has a good text (amazon.com/Number-Fields-Universitext-Daniel-Marcus/dp/…) as well as Samuel (amazon.com/Algebraic-Theory-Numbers-Translated-Mathematics/dp/…) – jspecter Sep 20 '11 at 14:45
• @jspecter I was about to recommend the very same! With a few more words your comment would make a good answer. – Dylan Moreland Sep 20 '11 at 15:07
• It would help if you revealed more about your goals. For example, do you plan to go on to do research in number theory or, is your aim to obtain a solid background for applications such as cryptography? – Bill Dubuque Sep 20 '11 at 17:55
• Research in Number theory. Actually, I'm interested in the Langlands program. – Mohan Sep 21 '11 at 8:18

## 7 Answers

I would recommend Ireland and Rosen's text Classical Introduction to Modern Number Theory. The prerequisites for the first 10 or so chapters are minimal (first semester undergraduate algebra course).

• Although very good, this is an elementary number theory textbook with one chapter ( out of 20 ) dedicated to ANT. – nilo de roock Oct 7 '15 at 15:56

I'm a big fan of Murty and Esmonde's "Problems in algebraic number theory", which develops the basic theory through a series of problems (with the answers in the back). Another nice sources in Milne's notes on algebraic number theory, available on his website here.

• Milne's notes have answers at the end as well. – Dylan Moreland Sep 20 '11 at 15:08

I would recommend Stewart and Tall's Algebraic Number Theory and Fermat's Last Theorem for an introduction with minimal prerequisites. For example you don't need to know any module theory at all and all that is needed is a basic abstract algebra course (assuming it covers some ring and field theory). But again the first chapter gives a quick review of the basic material from abstract algebra that is used later on.

I have used this book when I took an introductory course in Algebraic Number Theory and the experience was really good. The book has many examples and the pace is not too fast.

Most other books I have seen rely more heavily on module theory to make the exposition more general but since probably you don't have that background yet then I guess that this book may very well suit you perfectly.

• Add Stewart's book on Galois Theory to the 'minimal' requirements. The book is full of references to it. Both books are too sketchy to really grasp the topic from it. – nilo de roock Oct 7 '15 at 15:57
• I second ndroock1. Stewart and Tall's Algebraic Number Theory and Fermat's Last Theorem is supposed to be an introduction to the subject, but it has several logical gaps that beginners may find hard to fill in. I would choose Alaca and Williams instead. – eltonjohn Jan 23 '16 at 1:37

I liked Algebraic Theory of Numbers by Pierre Samuel.

I like Algebraic Number Theory, Second Edition by Richard A. Mollin. It is simple and best book to self study.

Paulo Ribenboim, A Classical Introduction to Algebraic Numbers, Springer 2001 https://www.goodreads.com/book/show/1873959.Classical_Theory_of_Algebraic_Numbers " The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics. "

I recommend Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory by Harold M. Edwards.

I quote:

This introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization. The more elementary topics, such as Euler's proof of the impossibilty of $x^3 +y^3 = z^3$, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's theory to quadratic integers and relates this to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.