Math 627 (Section 600) – Fall 2016

Algebraic Number Theory

MWF 12:40 – 1:30
BLOC 160


Course Description:

Topics of Study: This course will be an introduction to the study of algebraic numbers and algebraic integers. In number theory the key motivating problem is to understand the basic arithmetic of the integers. Algebraic number theory is the study of generalizations of integers to other domains, especially to number fields, ie, finite algebraic extensions of Q. Interesting problems arise in the study of rings of algebraic integers that shed light on many basic number theory problems. The course will cover the following topics as time permits:

  • algebraic numbers and algebraic integers
  • number fields and rings of algebraic integers
  • quadratic and cyclotomic fields
  • Dedekind rings
  • factorization of algebraic integers and unique factorization of ideals
  • Minkowski's Theorem and the geometry of numbers
  • ideal classes and the finiteness of the class number
  • Dirichlet's unit theorem
  • splitting of prime ideals in extensions and Artin reciprocity
  • Dedekind zeta function and class number formulas

Prerequisites: Students should be familiar with the topics covered in a graduate course in Algebra (Math 653), including standard results on groups, rings, fields, and vector spaces. No previous background in number theory is necessary.

Students interested in doing research in Number Theory are especially encouraged to attend this course.

Course Information:

Instructor: Matthew Papanikolas

Office Hours: Mon. 1:30-2:30, Wed. 10:30-11:30; also by appointment

Office: Blocker 641H


Textbook: Number Fields, by Daniel A. Marcus, Springer, 1977, ISBN 0-387-90279-1.

Prerequisites: Math 653 (groups, rings, fields, and vector spaces); or equivalents

Course Webpage:

Course Work and Grades: There will be regular homework assignments. These will serve as the basis for grades in the course.

Additional References: Algebraic Theory of Numbers, by Pierre Samuel, Dover, 2008, ISBN 978-0-486-46666-8.

Algebraic Number Theory, by Jürgen Neukirch, Springer-Verlag, 1999, ISBN 3-540-65399-6.

Algebraic Number Theory, by Serge Lang, Springer-Verlag, 1986, ISBN 0-387-96375-8.

Page maintained by Matt Papanikolas, Dept. of Mathematics, Texas A&M University.