TENTATIVE SYLLABUS
MATH 571
Introduction to Number Theory
Topics:
This will be an introductory graduate course, designed to cover the main
prerequisites for our further graduate course offerings, such as Iwaniec's
usual graduate courses. I will start with algebraic topics, but
present
them with an analytic perspective. I will then turn to analytic
techniques
proper, such as the proof of the prime number theorem. If time
permits,
we will discuss the basics of modular and maass forms, and perhaps
elliptic
curves. My overall aim is to cover many topics and describe the role
of the key ideas involved.
1. Elementary Number Theory
2. Euclidean Rings
3. Algebraic Numbers and Integers
4. Integral Bases
5. Dedekind Domains
6. The Ideal Class Group
7. Quadratic Reciprocity
8. The Structure of Units
9. Higher Reciprocity Laws
10. Zeta Functions
11. Prime Number Theorem
12. Modular Forms
13. Maass Forms
14. Elliptic Curves
The recommend text (below) is a novel book that is broken in to two
parts.
In the first part, a brief discussion of the topic is given, followed by
numerous problems, many of which are fundamental examples. Detailed
solutions are given in the second half. I believe this will serve
our
course better than some traditional textbooks because it is very up
to date, and almost perfectly aligned with the topics that are important
to students pursuing graduate work in number theory here at Rutgers.
I will cover the material of this book, and then add more from other
sources.
Incidentally, Murty has a separate volume, Springer Graduate Text #206,
entitled
"Problems in Analytic Number Theory" which is another excellent
reference.
Recommended Text Book: M.
Ram Murty and Jody Esmonde, Problems in
Algebraic Number Theory, Springer-Verlag Graduate Texts in
Mathematics,
volume 190.
Other reference books: Murty, Problems
in Analytic Number Theory, Springer GTM, volume 206.
Lang, Algebraic Number Theory,
Springer
GTM, volume 110
Ireland and Rosen, A Classical
Introduction
to Modern Number Theory, GTM, volume 84.
Suggested HW problems:
2.1.8, 2.2.1, 2.3.1, 2.3.2,
2.3.4, 2.4.6, 2.4,7, 2.5.2, 2.5.8,
3.13, 3.3.7, 4.2.7, 4.2.8,
4.5.3, 4.5.4, 4.5.5, 4.5.6, 4.5.8, 4.5.11, 4.5.12, 4.5.21, 4.5.23,
4.5.26
5.1.1, 5.1.3, 5.3.1, 5.3.12, 5.3.14, 5.3.17
Fall
2003
