Algebraic Number Theory

Instructor

Prof. Paul Gunnells, LGRT 1115L, 413.545.6009, gunnells at math dot umass dot edu.

Office Hours

Tuesdays and Thursdays, 8:30-9:30, and by appointment.

Overview

Resources

Textbook

Notes by James Milne. Freely available online. He requests that you only print out one copy for your personal use.

Other books

• Cassels, Frohlich, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). A standard reference, with expositions of many topics, including local/global fields, cohomology of groups, class field theory, towers of class fields, Hecke L-functions and their functional equations, and a fun historical section. Very challenging for the novice, but it's all there. Sadly still out of print.
• Borevich, A. I.; Shafarevich, I. R. Number theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20 Academic Press, New York-London 1966. A classic. Very hands on. Lots of examples.
• Manin, Yuri Ivanovic; Panchishkin, Alexei A. Introduction to modern number theory. Fundamental problems, ideas and theories. Translated from the Russian. Second edition. Encyclopaedia of Mathematical Sciences, 49. Springer-Verlag, Berlin, 2005. Many precise definitions, but few complete proofs. Gives a wide overview of the subject.
• Koch, H. Algebraic number theory. Translated from the 1988 Russian edition. Reprint of the 1992 translation. Springer-Verlag, Berlin, 1997. The companion volume to Manin-Panchishkin. Similar in style. An excellent way to see the whole subject in the large without getting bogged down in the details.
• Frohlich, A.; Taylor, M. J. Algebraic number theory. Cambridge Studies in Advanced Mathematics, 27. Cambridge University Press, Cambridge, 1993. Emphasizes the role of valuations.
• Neukirch, Jürgen Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999. Excellent book. Lots of material, including class field theory. Nice treatment of number fields from the point of view of Arakelov geometry. Zeta and L-functions too (the approach complements the material in Cassels-Frohlich). Full details of just about everything.
• Lang, Serge Algebraic number theory. Second edition. Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994. A classic. Unique tone.
• Marcus, Number Fields. Springer-Verlag, New York, 1977. Another classic. Very hands on, with many detailed examples and exercises. Highly recommended by certain real number theorists (like Farshid).
• Rodriguez Villegas, Fernando. Experimental number theory. Oxford Graduate Texts in Mathematics, 13. Oxford University Press, Oxford, 2007. Shows how to use computation to explore number theory and to formulate conjectures. Similar in spirit to many examples done in class (indeed, I learned tons of number theory from Fernando by having personal demonstrations of these ideas at various conferences and cafes). Very inspiring. Highly recommended.

Online references

Algebraic number theory is a central topic; there are many freely available sets of lecture notes floating around, in addition to Milne's. Here are a few I know about. I haven't read them, so can't give comments.

• From our very own Tom Weston.
• C.L.Chai's notes.
• Willaim Stein's notes.
• Many, many more. Not all of these are about algebraic number theory, but I think you get the idea.

Software

• Pari-GP is the standard tool for people wanting to do computations in number theory. Free, easy to install, easy to use. I use it all the time for quick computations, even those having nothing to do with number theory. Under active development with a large user group.
• KANT/KASH is another program that has a lot of the functionality of Pari, and offers more functionality in some other areas (like in dealing with finite fields and algebraic function fields). Free, easy to install, easy to use. Not sure if it's still in active development.
• SAGE is a free computational algebra system that includes Pari-GP as a subset (as well as many other free software packages).

Problems

Here are some exercises so that you can put some of the lecture material into practice. I will not collect and grade these, but I'm certainly willing to discuss them. Some of the problems will need a computer; I recommend Pari-GP.

• I

Code

Here are some GP scripts that were developed/demonstrated in class. You might also want to have a look at the Bordeaux database of number fields of low degree for some examples to play with.

• uniq.gp. Removes duplicate elements from a sorted vector.
• factsig.gp. Factors a polynomial mod p and returns the degrees of the irreducible factors.
• autorder.gp. Computes the order of a Galois automorphism.
• frob.gp. Computes the Frobenius conjugacy class in a Galois extension.

Grading

The grades for this course will (yet again) be based on a final paper. This will be an expository article of no less than five and no more than ten pages that you'll prepare on a topic related to material in the course and your own interests. My goal is to simulate as accurately as possible the experience of writing an original research paper. Here's how it will work:

• When there are six weeks (or so) left in the term, I'll talk about a possible topic with you. I'll try to match topics to students based on their own interests, and will give you pointers to the literature.
• You will research these topics outside of class and start to prepare a draft. You can ask me for advice if you need help. As an example of what kind of writing I have in mind, you can consult my modular forms and singular spaces TWIGS talks. These papers are on the short side, since the talks were only an hour. You could pretend that you have two hours to give a lecture, and write up notes from that.
• A draft of your paper should be completed within three weeks (a deadline will be given later), which you will submit to me. The draft should be in some flavor of TeX (I use AMS-LaTeX); in particular this will be a good chance for you to learn TeX if you don't already know it. I will read these drafts and make comments.
• You'll get the drafts back as soon as possible with lots of comments, which you should not take personally. You will address the comments and prepare a new draft. I will give you even more nitpicky and annoying comments, and so forth. Experience shows that it'll require a few iterations of this before everyone (i.e. me) will be happy. Then you will prepare and submit a final version.

Any kind of writing is challenging, and writing mathematics poses its own challenges. It's rare for a graduate student to get detailed feedback on writing before his or her thesis, but I feel that such feedback would have been extremely helpful to me. This should be a good opportunity for you to get some. Also, students in the past have been very enthusiastic about this writing assignment (admittedly, they're enthusiastic after they're finished), even though it's a lot of work and is not without a certain amount of pain. In fact, I've never had a student who did it who didn't have a good experience. You will (probably) thank me some day.

For more information about writing and mathematical writing in particular, you can consult the following:

• Mathematical writing, by Don Knuth.
• Milne's website. Take a look at the Tips for Authors link.
• J.-P. Serre's Hints on mathematical style. He's a master of mathematical exposition.
• A primer on using LaTeX can be found here.
• MIT has a writing requirement for its undergraduates, and Steve Kleiman has prepared an excellent introduction to writing short mathematical papers. There are also some notes available about using LaTeX, although a lot of features are peculiar to MIT's style files.