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, GTM, volume 84.

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.

Description: 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.

Topics:

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
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