Algebraic Number Theory
Mathematics 605A: spring 2016
Lecture: MWF 10:00-11:00, Engineering E206.
Prerequisite: Math 566 or permission of professor.
Homework and reading list:
Detailed information on homework
Spring 2016 version or
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 deep and 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 aspects. Here are some of the themes of the course.
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 rings of integers in number fields. The proof relies on Minkowski's theorem on the geometry of lattices in the plane.
3) Kummer theory, class groups, abelian extensions of number fields, applications to Fermat's last theorem.
4) Lattice based cryptosystems: these are the only cryptosystems currently immune to quantum attacks
The course grade will be based on 50% homework, 20% presentations, 30% final project.
See syllabus for long list of references.
Here are links to several on-line ones:
Milne: Algebraic Number Theory
Stein: Brief Introduction to Classical and Adelic Algebraic Number Theory
Ash: A Course in Algebraic Number Theory
Ogglier: Introduction to Algebraic Number Theory
Marcus: Algebraic number fields
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.