2018/2019
Научно-исследовательский семинар "Алгебра и арифметика"
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Преподаватели:
Жгун Владимир Сергеевич
Язык:
английский
Кредиты:
5
Контактные часы:
60
Course Syllabus
Abstract
We plan to start from the algebraic properties of integer numbers, arithmetic of residues and basic properties of polynomials: such as Chinese remainder theorem, little Ferma's theorem, Wilson's lemma, quadratic residues. This will give us motivation for introducing more general notions in the group theory, commutative and non-commutative algebra. In particular we shall also study basic properties of finite groups such as cosets, normal, nilpotent and solvable subgroups, Sylow theorems, basic notions of commutative algebra: such as ideals, modules, maximal and prime ideals, localization, and basic notions of non-commutative algebra.
Learning Objectives
- The aim of the course is to give an introduction to basic notions of algebra and number theory.
Expected Learning Outcomes
- Successful participants will master fundamental tools from algebra and number theory.
Course Contents
- Basic notions of integer numbers and residues
- Chinese remainder theorem, little Ferma's theorem, Wilson's lemma
- Quadratic residues. Gauss reciprocity law.
- Basic notions of group theory. Cosets, normal, nilpotent and solvable subgroups.
- Group actions. Orbits, stabilizers, normalizers, conjugacy classes. Burside formula.
- Sylow theorems
- Basic notions of commutative algebra: rings, fields, algebras, ideals, modules.
- Properties of finite fields.
- Nilpotence, radicals, maximal and prime ideals, localization.
- Basic notions of non-commutative algebra. Structure theory for non-commutative algebras.
Assessment Elements
- Cumulative gradeA cumulative grade is proportional to number of tasks solved
- Final exam
Bibliography
Recommended Core Bibliography
- Ash, R. B. (2007). Basic Abstract Algebra : For Graduate Students and Advanced Undergraduates. Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152113
Recommended Additional Bibliography
- Axler, S. J. (1997). Linear Algebra Done Right (Vol. 2nd ed). New York: Springer Science & Business Media. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=104527
- Körner, T. W. (2013). Vectors, Pure and Applied : A General Introduction to Linear Algebra. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=508905
- Robinson, D. J. S. (2015). Abstract Algebra : An Introduction with Applications (Vol. 2nd ed). Berlin/Boston: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1000456
- Singh, K. (2014). Linear Algebra : Step by Step. Oxford: OUP Oxford. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=652224
- Weyl, H. (1998). Algebraic Theory of Numbers. (AM-1), Volume 1. Princeton, N. J: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1204558