## Algebraic Number Theory (Math 8630), Spring
2016
Instructor: Andrew Obus
email: obus [at]
virginia.edu
office: Kerchof Hall 208 |

Problem session

There will be a (roughly) biweekly one-hour problem session where students will present homework solutions on the board, and where we can discuss the solutions. The meeting time is Wednesday 5-6. Room is TBA.

Textbook

*Algebraic Number Theory*, by Jürgen Neukirch. There
should also be a copy on reserve in the
library. This book is a "modern classic" in the field, in that it
takes great care to keep an algebro-geometric perspective throughout
(while still being accessible to those with no background in algebraic
geometry). The book contains much more material than we will be
able to study, and will be a good reference for future work. It
also contains lots of exercises. There is a bit of nonstandard
terminology --- I will point this out as we encounter it.

- Local Fields (Corps Locaux),
by Jean-Pierre Serre. A more streamlined exposition of local class
field theory, including the basic local material we will cover.
Has a few topics not covered in Neukirch.

- Number Fields,
by Daniel Marcus. This book is written at a somewhat lower level
than Neukirch, and deals only with the global picture. However,
it is nicely written and has a huge
number of computational exercises. Good for practice!

- Algebraic Number Fields,
by Gerald J. Janusz. I'm not so familiar with this book, but
people seem to like it. Perhaps "Neukirch-lite" is the best
description. Does topics in a somewhat different order.

- Algebraic Number Theory,
by Serge Lang. Covers a huge amount in a relatively short
book. Includes analytic theory as well.

- Algebraic Number Theory, by J. W. S. Cassels and A. Fröhlich. Covers similar material to Lang. Comprised of separate chapters by many of the giants in the field. Contains a reproduction of Tate's thesis on functional equations of L-functions.
- Basic Number Theory, by
André Weil. Similar material to Cassels
and Fröhlich, but using much more topology and less Galois
cohomology. Don't be fooled by the title --- this is a difficult
book!

We will cover most of the material in chapters I and II of Neukirch. This includes all of the standard results (finiteness of the ideal class group, structure of the unit group, Hensel's lemma, ramification theory). We will not do this in order, and will spend some time on the local (p-adic) theory before jumping into the global theory (algebraic number fields). I find that thinking of number fields in terms of their localizations lends itself better to retention. If time permits, we will cover a few more topics (in particular, adeles and idèles).

The prerequisite for this course is Algebra III. Most of the material from that course is not so essential, but you should know what a tensor product is. If you have only had Algebra I and II and want to take the class, this should be OK. The main material from Algebra I and II on rings, modules, and fields is indispensable; I do not recommend trying to take the course concurrently with Algebra II. Occasionally, I will use commutative algebra concepts such as integrality, localization, completion, and valuations. These will be recalled as needed.

Tu 11-12, W 10-11. Kerchof Hall 208 (my office). If the times do not work for you, please make an appointment with me.

There will be 6 homework assignments over the course of the semester. Some of the problems will be difficult! The homework assignments will be posted on Collab.

Disabilities

All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.

If you have (anonymous) comments for me about teaching style or anything related to the course, you can make them on the Collab page for the course.