# Text for Algebraic Number Theory

I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some commutative algebra, some homological algebra, and some K-theory. The texts I am now considering are

(1) Frohlich and Taylor, Algebraic Number Theory,

(2) Janusz, Algebraic Number Fields, and

(3) Milne, Algebraic Number Theory (online).

I am leaning toward (1): it seems very well written and has sections on cubic, biquadratic and sextic fields as well as quadratic and cyclotomic fields. I do know that when you get into a course you discover drawbacks that aren't apparent when you're reading the text before the course begins. Frohlich and Taylor is designed for a year long course and I only have one semester, so will need to do some selecting. I regret not using any homological techniques, also.

I would appreciate suggestions, either about these or other texts.

• Janusz is my favorite among these three. You should also look at Neukirch's book. – Andy Putman Jan 21 '14 at 20:22
• I probably should have mentioned Neukirch as well, but it seemed a bit like taking a sledge hammer to a nail! I fear that I don't have the dexterity to handle it well. On the other hand, i still remember the delight I felt when I first sat down with Neukirch. – Robert Bruner Jan 21 '14 at 20:24
• Neukirch is inspiring and does a lot of stuff, though I agree that it takes a firm hand to teach a good course from it. Strangely enough, I actually prefer the line-by-line writing in Janusz, though it is less "exciting". – Andy Putman Jan 21 '14 at 21:07
• If I were one of the students, I would like to enjoy with Serre, Local Fields (completing the standard missing theorems with Neukirch). But probably, as you say, it is more prudent to choose Frohlich-Taylor. – Vinteuil Jan 21 '14 at 21:13

There is a short (130 pages), well-written and dense book on the subject : Algebraic Theory of Numbers, by Pierre Samuel ( translated from French by Allan J. Silberger ). Very good for a semester course.

I'm a big fan of Milne's, however, I'm finding Froehlich-Taylor currently the most balanced mid-level text book available. That being said, if I were you I'd have a good look at H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, 2001, which manages to be so wonderfully efficient, while at the same time, quite like F-T, it also brings plenty of relevant examples (a whole chapter 3).

Janusz certainly merits a place as a classic of class field theory, but is overall perhaps not as student-friendly as the other two. On that note, Algebraic Numbers by Paulo Ribenboim (Wiley-Interscience 1972) is extremely gentle with the reader and driven by concrete problems.

Best of luck with your course! Raphael

In my opinion, the book "Algebraic number theory" by Serge Lang deserves to be mentioned in this list, as well.

My favorite textbook on the subject is "Number fields" by Marcus. Things are done in a very explicit way, there are hundreds of exercises, and I've usually found that the proofs of the classical theorems (finiteness of the class number, structure of the units, ...) are done in a way that I really like.

• I wish this book was typeset in $\TeX$. It's great otherwise. – lhf Mar 12 '14 at 2:40

I think the book Algebraic number theory by Helmut Koch should be mentioned too, together with his book Number Theory: Algebraic Numbers and Functions. For an overview and a discussion see the talk given on The Mathematical Work of Helmut Koch. The online lecture notes of Milne are also excellent, in my opinion, and contain the theory together with many examples.

Olivier Bordellès, Arithmetic Tales, Springer.

It's not entirely focused on algebraic number theory which is the subject of the last part of the book, but almost no prerequisite is needed. The author introduces all necessary elements about Galois theory and whatever might be needed to start studying algebraic number theory. This book is a revised and augmented English version of his former book "Thèmes d'arithmétique" which is already a very nice work focused on elementary and analytic number theory, with many exercises, and that suits people from "bac +1 level" to master's degree (I don't know the American educational system, so I stick with the French one, sorry).