Научно-исследовательский семинар "Алгебра и арифметика"

2018/2019

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

Full 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 grade
A cumulative grade is proportional to number of tasks solved
Final exam

Interim Assessment

Interim assessment (2 module)
0.3 * Cumulative grade + 0.7 * 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

Course Syllabus