MATH 620: Algebraic Number Theory (Fall 1999)

Title: Algebraic Number Theory
Course web site: http://www.math.umd.edu/~jmr/620/
Meeting times: MWF, 1:00pm-1:50pm (MTH 2300)
Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. His office hours are tentatively scheduled for Tuesday and Thirsday afternoons 1-2PM.
Text: Algebraic Number Fields, Second Edition, by Gerald J. Janusz, Graduate Studies in Mathematics, American Mathematical Society. I expect to cover chapters 1-3, along with some applications to "classical" number theory (e.g., "Pell's equation" and quadratic reciprocity). We may touch on parts of chapters 4-6 if time permits.
Prerequisite: Graduate-level abstract algebra (MATH 600-601)
Catalog description: Algebraic numbers and algebraic integers, algebraic number fields of finite degree, ideals and units, fundamental theorem of algebraic number theory, theory of residue classes, Minkowski's theorem on linear forms, class numbers, Dirichlet's theorem on units, relative algebraic number fields, decomposition group, inertia group and ramification group of prime ideals with respect to a relatively Galois extension.

Course Description:

This course is a basic introduction to algebraic number theory. It will emphasize the following topics:

number fields, local fields, and their extensions
rings of algebraic integers, basics of Dedekind domains
finiteness of the class number
the Dirichlet unit theorem and applications

If possible we will try to do somewhat more on quadratic number fields and give an introduction to zeta-functions and L-functions and to some ideas of class field theory.

Course Requirements:

Homework will be collected and graded regularly. In addition, there will be a take-home final exam at the end of the semester, due on Monday, December 20. Homework will count for 2/3 of the grade, the exam for the other 1/3.