Graduate Number Theory
Mathematics 676: fall 2010
Lecture: MWF 2:00-2:00, Engineering E104. code 63412
Prerequisite: Math 566 or permission of professor.
Detailed information on homework
Official proposed version
Version from 2007 including references
The study of number theory originated in ancient civilizations
such as those of China and India and was developed in great depth in Europe
in the 17th and 18th centuries.
Number theory is known for having problems that are easy to state yet which
can only be solved using complicated structures.
For example, it took 300 years to find a complete proof of Fermat's Last Theorem.
Number theory is a subject that's intertwined with group theory, algebraic
geometry, combinatorics, and complex analysis. It's become popular recently
because of its applications to coding theory and cryptography.
Number theory is a vast subject. In this course, we will emphasize its algebraic and geometric aspects. Here are some of the possible themes of the course. In the first week of class, we will have an introduction to these topics and choose which subset of them to cover.
1) Reciprocity Laws: the quadratic reciprocity law (which has over 100 proofs) tells you whether or not a number is a square modulo a prime.
2) Ideal Factorization: this topic helps you measure the failure of unique factorization in quadratic integer rings. The proof relies on Minkowski's theorem on the geometry of lattices in the plane. These lattices are used in recent cryptosystems.
3) Riemann zeta function: this function measures how primes are distributed among the integers. The Clay Mathematics Institute is offering $1,000,000 for the the solution to the Riemann hypothesis.
4) Diophantine equations: this topic is about solutions to equations with coordinates in fields like Q or F_p. It includes the topic of elliptic curves (again with applications to cryptography) and Fermat's Last Theorem. Some of the best data-transfer codes rely on this topic.
The course grade will be based 60% on homework and 40% on final project (15% presentation and 25% written).
The project is an opportunity to learn more about a topic in number theory that interests you or will be relevant for your future graduate work. It gives us a chance to hear about important ideas which we will not have time to cover in class. It is also a good opportunity to develop more skill at writing and speaking on mathematics.
Help: Help is always available if you have trouble with homework or lecture material. If your classmates can't answer your question, come ask me! Office hours will be (TBA) or are available by appointment.