Instructor: Dr. Farshid Hajir.    Office: Lederle Graduate Research Tower 1118.    Phone: 545-6015. e-mail: hajir@math.umass.edu
I encourage you to use the email address above to send me questions about the homework or to set up an appointment.

Description: This course will cover the basic theory of algebraic numbers, including an introduction to class field theory. We start with a very concrete problem: given a positive integer n, what numbers can be expressed as a square plus n times a square? We study some commutative algebra (integral closure, Dedekind domains) and give the fundamental properties of the ring of algebraic integers of a finite extension of the rational numbers (finiteness of class group, finite generation of the group of units). We will concentrate on the class field theory of imaginary quadratic fields, i.e. of Galois extensions with abelian Galois group over such fields, and their connections with torsion points on elliptic curves, modular forms, and L-functions. There will be bi-weekly problem assignments. You can choose to complete a final project or to take a final exam. This choice should be made by 10/15. A list of possible projects will be made available shortly.

Text: David Cox, Primes of the form x^2+ny^2, Wiley 1989. I also recommend Daniel Marcus, Number Fields, Springer. There are many other texts; I'd be happy to discuss them with you. Our own Tom Weston has some excellent (free!) unpublished lecture notes which I recommend highly for certain topics not really covered in Cox's book. I will also occasionally put some lecture notes here.

Prerequisites: Math 611 and 612 or equivalent. If you have not taken M611, M612, please speak to me about whether this is the right class for you.

Meeting Time and Place: Tues, Th 9:30, LGRT 1334.

Office Hours: During the first two weeks, my office hours will be Tues 1:30-2:30, Wed 2:00-3:00. After the first two weeks, my permanent office hours will be announced in class and posted on my website. You are always welcome to set up an appointment to see me by e-mail or phone.

Homework Problems: They'll be listed separately at http://www.math.umass.edu/~hajir/m713/m713hw.html.

FinalProject: Details are contained in the Final Project Handout.

Exams: As mentioned above, you have until 10/15/05 to choose an optional final in lieu of a final project.

Grading: Homework and class participation (60%), Final Exam or Final Project (40%).

Homework, Attendance, and Collaboration: Attendance and class participation are important, as are the homework assignments. You may occasionally be asked to present a homework problem on the board. I recommend that you WORK WITH YOUR FELLOW STUDENTS IN GROUPS!! If you are stuck on a problem and seek help from an instructor or a fellow student, you owe it to yourself to aim for an understanding of the concepts and ideas that come up in the discussion (do not just memorize the series of steps leading to the solution). Then, go home and reconstruct the argument for yourself in the privacy of your own brain, to make sure you are not merely reproducing mindlessly something you have not thought through. It is of paramount importance that you do the write-up completely independently. Failure to do so will be cause for disciplinary action.