2019/2020
Научно-исследовательский семинар "Алгебраическая теория чисел"
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Преподаватели:
Ровинский Марат Зефирович
Язык:
английский
Кредиты:
6
Контактные часы:
72
Course Syllabus
Abstract
The goal of the course is to introduce some basic notions and results of related to the algebraic extensions of the field of rational numbers. One of the objectives will be a relatively explicit construction of abelian extensions of p-adic fields using formal groups (Lubin–Tate theory). If time permits, the Galois groups of abelian extensions of number fields (Global class field theory) and related Artin reciprocity law (generalizing Gauß’ quadratic reciprocity) will also be discussed.
Expected Learning Outcomes
- You will be able to construct the abelian extensions of the local and of certain global fields.
Course Contents
- Global (e.g., number) fields, their invariants and topologies on them
- Local fields (e.g., p-adic numbers), ramification theory, their multiplicative structure
- Relations between local and global properties (Hasse principle), Minkowski–Hasse theorem
- Commutative algebra of Dedekind domains; finiteness theorems for units and class groups
- Hilbert symbol and quadratic forms over p-adic fields
- Formal groups and local class field theory
- Tate cohomology, norm residue symbol and «abstract» class field theory
- Simple algebras and Brauer groups; Brauer groups of local fields; Brauer groups of global fields
- Global class field theory; reciprocity laws
Interim Assessment
- Interim assessment (4 module)min[10,20/3((ratio of solved problem of the problem sets) + (ratio of solved problem of the final exam))].
Bibliography
Recommended Core Bibliography
- J. S. Milne. (2009). Algebraic Number Theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.CB7FD32F
Recommended Additional Bibliography
- J. S. Milne. (2008). Class Field Theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.3A8A3078
- Yoshida, T. (2006). Local class field theory via Lubin-Tate theory. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.math%2f0606108