Prerequisites: We will assume a working knowledge of first year graduate level algebra. *Permission of instructor required for students not enrolled in the mathematics Ph.D. program.

Description:

Topics: This will be a introductory course in Algebraic Number Theory. The subject matter of the course should be useful to students in areas of algebra and discrete mathematics, which often have a number theoretic component to problems, as well as students in number theory and algebraic geometry. Basic properties of number fields (field extensions of the rational numbers of finite degree) will be introduced -- rings of integers, ideal classes, units groups, zeta functions, adele ring, group of ideles. The relation of these invariants to Diophantine problems of finding rational solutions to collections of polynomial equations will be developed. Examples such as quadratic and cyclotomic fields will be studied. Galois extensions of number fields, Chebotarev density theorem and L-functions will be introduced.

1. Number fields, lattices and rings of integers
2. Dedekind domains and their ideals and modules
3. Ideal class groups and Class number
4. Zeta functions of number fields
5. Quadratic fields and binary forms
6. Cyclotomic fields and Gauss sums
7. Diophantine problems and algebraic number theory
8. Algorithms in number theory
9. Adeles and Ideles of number fields
10.Chebotarev Density Theorem