# Algebraic Number Theory

## Instructor

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

TBA.

## Overview

An algebraic number field is a field obtained by adjoining to the rational numbers the roots of an irreducible rational polynomial. Algebraic number theory is the study of properies of such fields. This course will cover the basics of algebraic number theory, with topics to be studied possibly including the following: number fields, rings of integers, factorization in Dedekind domains, class numbers and class groups, units in rings of integers, valuations and local fields, and zeta- and L-functions.

## Resources

### Textbook

Algebraic Number Theory by James Milne. Freely available online. This is a very polished textbook that covers all the main topics in algebraic number theory. The only serious omission is zeta and L-functions, but they are discussed in his notes on class field theory. (One might argue, as Chevalley did, that such objects should remain unmentioned in algebraic number theory and class field theory, but we will adopt a more inclusive stance.)

### 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. Recently back in print. Unsophisticated web searching finds this.
• 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 number theorists for self-study.
• 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.

### Software

• Pari-GP is one of the standard tools for 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.
• SAGE is a free computational algebra system that includes Pari-GP as a subset (as well as many other free software packages). Under active development with a large user group.
• Magma is software package designed for computations in algebra, number theory, algebraic geometry and algebraic combinatorics. The philosophy of this system is rather different from that of Pari-GP. Like SAGE, it's much more "object-oriented." I don't use it much myself but know many that do and swear by it. 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.

## Problems

Here is the list of problems (it will be updated every so often). Some problems might benefit from computer assistance; I recommend Pari-GP, SAGE, or Magma (the latter is not free but should be freely available to you as a grad student in our department).

## Code

Here are some GP scripts that have been used in prior incarnations of this 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.