189-726A: Topics in Number Theory I
Algebraic Number Theory
Professor: Henri Darmon
Time: MWF 10:30-11:30.
Room: Burnside 1205.
Required Text: Introduction to Algebraic Number Theory, by
J. Frolich and M. Taylor, Cambridge University Press.
Syllabus: The theory of algebraic number fields developped by
Dedekind, Dirichlet, and Kummer is one of the crowning
achievements of 19th century mathematics.
Our goal is to give an introduction to
the subject, covering most of the
fundamental results known at the end of the 19th century, but not
the class field theory that was developped in the first half
of the 20th century.
Topics will include:
If time permits, I might also discuss Artin L-functions, or some
rudiments on elliptic curves.
- Dedekind domains, gcd, and unique factorization;
- Finiteness of the class group;
- Dirichlet's Unit theorem;
- Quadratic fields and cyclotomic fields;
- Dedekind zeta-functions and Dirichlet's class number formula.
Grading Scheme: To be determined.