Text:There will be no required text for this course. Diverse texts will
be on reserve and notes will be distributed for certain topics .
Other reference books:
M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory,
Springer-Verlag Graduate Texts in Mathematics, volume 190.
Lang, Algebraic Number Theory, Springer GTM, volume 110
Ireland and Rosen, A Classical Introduction to Modern Number Theory,
Course Format: There will be periodic problem
assignments and term projects involving algebraic number theory which
will investigate applications and extensions of the material.
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.
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. The basic
invariants of field extensions of finite degree over the rational
field (so-called number fields) will be introduced --- ring of
integers, class number, units group, zeta functions, adele rings and
group of ideles. The relation of these abstract invariants to the
problem of solving polynomial equations in integers will be developed.
Special examples of number fields such as quadratic and cyclotomic
fields which have a rich structure will be used to illuminate the
theoretical aspects. Galois extensions of number fields will studied
along with the Chebotarev density theorem.
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