Syllabus for Math 428. Spring, 2009.

Professor: Scott Ahlgren
Email: ahlgren@math.uiuc.edu
Phone: 244-1738
Office: 302 Altgeld Hall

Course Description: This is a course in number theory at the honors undergraduate level. We will prove some important theorems using a variety of techniques. Topics include:

Finite fields. Equations over finite fields. Applications (quadratic reciprocity, for example).
Methods from analysis. We will prove Dirichlet's famous theorem on primes in arithmetic progression.
p-adic methods. The goal here is the Hasse-Minkowski theorem, which characterizes those quadratic forms over the rationals which have a zero.

This list may be modified or changed depending on the background and the ambitions of the students in the class.

Text. A classical introduction to modern number theory, by Ireland and Rosen.

This is a great introduction to many aspects of the subject--it is a book that you will want to keep. We will cover chapters 1-8 and chapter 16, as well as some material which is not in the text.

Webpage. This syllabus can be found on the course webpage: http://www.math.uiuc.edu/~ahlgren/math428sp09/math428.html

I will not be posting assignment or announcements on the webpage since the class is small.

Classes. Monday, Wednesday, Friday, 10-10:50 am. 145 Altgeld Hall.

Grading. Your course grade will be computed as follows:

Problems: 80 points
Project: 20 points
Total: 100 points

Attendance. I expect students to be in class. There may be some in-class aspect to the "Problems" portion of the grade.

Problems sets. You will spend a lot of time thinking about problems, and there will be regular problem sets.

Project. You will research a topic related to the course, prepare a summary, and give a talk to the class. The talks will occur near the end of the semester, and perhaps during the final exam period.

Office hours. By appointment. Email me or talk with me after class to set up a time to talk.

Other books. There are many books in the mathematics library on the topics in this course, and many online resources. You are encouraged to explore these--let me know if you want suggestions, or if you find a book which is especially interesting or helpful.

Final note. I am always open to productive suggestions about the course. Please feel free to contact me with any suggestions or concerns.

Best wishes for a productive and enjoyable semester!