Algebraic Number Theory (Math 8630), Spring 2016  

 
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
phone: 434-424-4930
website: http://people.virginia.edu/~aso9t/math8630s16.html

 

Lectures

TuTh 2:00-3:15, Monroe Hall 114.  Please ask questions if anything in lecture is unclear. Lectures will run the entire 75 minutes. Please show up on time!



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.

Other references you might enjoy for more-or-less standard material include:
If you want references for more advanced topics (Iwasawa theory, diophantine geometry, arithmetic dynamics, cohomology of number fields, modular forms, etc.), please talk to me!

Content

Algebraic Number Theory is the study of algebraic number fields (fields that are finite over Q).  Much of it arose from efforts to prove Fermat's Last Theorem.  Even though this is a question about Q itself, it turns out that answering it requires looking at "higher" algebraic number fields.  Many other problems in number theory (e.g., quadratic reciprocity, classifying the primes that are sums of two squares), are made easier from this perspective as well.  The structure of algebraic number fields is also fascinating in its own right, and offers many open questions.  For instance, is Z[\sqrt(d)] a unique factorization domain for infinitely many squarefree d?

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.

Office Hours

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.


Homework

You should be prepared to discuss/present homework solutions at the problem sessions, which will meet roughly every other week.  You are encouraged to work together on the homework, although I have found that this is more effective after having first spent some time thinking about the problems alone.  If you are officially enrolled in the class and have not yet passed all your qualifying exams (written and oral), then you will be responsible for handing in written homework.  I recommend TeXing solutions (I still reference my grad school homework solutions every once in a while).  If you want me to look over an answer (even if you are not obligated to hand the homework in), I am always happy to.

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.


Exams

There are no exams in this class.

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.

Comments

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.