Program
 Basic Number Theory Student No.： 50 Time： Mon/Wed 15:10-17:00, 07-01~08-21 Instructor： Tonghai Yang  [University of Wisconsin] Place： Conference Room 3，Floor 2, Jin Chun Yuan West Building Starting Date： 2013-7-1 Ending Date： 2013-8-21

The place of lecture from July 29 to August 2th will be changed to room A304 of Math department.

Description:

This short course will be taught in the form of lecturing and student discussion. Students are required to actively participating discussion in class. If you just prepare to sit in and listen to lectures without reading and studying yourself and discussing your thoughts with others, this course is not for you. The course’s intended audience is undergraduate students who might be interested in number theory but don’t know much about number theory.

We plan to use the proof of Catalan conjecture as an excuse to discuss basics of number fields, including Dedekind domains/factorization of primes, class groups and class numbers, class number formula, local fields, perhaps adeles/ideles, and L-functions. Catalan conjecture asserts that the only integral solutions to the equation $p^x –q^y =1$ are $p= \pm 3$, $q=2$, $x=2$, $y=3$, i.e., $(\pm 3)^2 – 3^3 =1$. It was proved by Mihailescu in 2002. The mail tools are basic number theory and a little class field theory.

Prerequisite: Knowing Abstract Algebra and a little elementary number theory will be very helpful

Reference:

J. Neukirch, Algebraic Number Theory

R. Schoof, Catalan’s Conjecture