2020/2021

# Research Seminar "Basic Algebra"

Category 'Best Course for New Knowledge and Skills'

Type:
Optional course (faculty)

Delivered by:
Faculty of Mathematics

Where:
Faculty of Mathematics

When:
1, 2 module

Instructors:
Alexander Pavlov

Language:
English

ECTS credits:
6

Contact hours:
60

### Course Syllabus

#### Abstract

The goal of the course is to learn basic concepts of abstract algebra. We start with a definition of an associative binary operation on a set and develop basics of the groups, rings, fields, vector spaces and algebras. Every abstract definition is illustrated by many examples. Special attention is paid to the quotients of groups and rings, as the most important construction in abstract algebra.

#### Learning Objectives

- Learn syntax and semantics of the group theory.
- Learn syntax and semantics of the rings theory with special attention to the univariate polynomial rings.
- Learn finite and algebraic extensions of fields.
- Learn splitting fields of polynomials and application to the finite field construction.
- Get acquainted with with algebras such as algebra of quaternions, tensor algebra and exterior algebra.

#### Expected Learning Outcomes

- Learn basics of the group theory.
- Learn basics of the ring theory.
- Learn basics of the field theory and see several examples of algebras.

#### Course Contents

- Introduction to Group Theory.Definitions and examples of groups. Normal subgroups and quotient groups. Isomorphism theorems. Group actions and their applications.
- Basics of Rings Theory.Definition and examples of commutative and non-commutative rings. Ideals and quotient rings. Modular arithmetic. Isomorphism theorems for rings. Prime and maximal ideals. Chinese remainder theorem. Univariate polynomial rings. Euclidean and principle ideal domains.
- Vector Spaces and AlgebrasReview of linear algebra: vector spaces, linear dependence, basis. Field extensions. Splitting fields. Finite fields. Definition of algebra and basic properties of algebras. Examples algebras: quaternions, tensor algebra and exterior algebra.

#### Bibliography

#### Recommended Core Bibliography

- Abstract algebra, Garrett, P. B., 2008
- 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
- Instructor Luís Finotti, Textbook D. Dummit, R. Foote, & Abstract Algebra. (n.d.). Math 551: Modern Algebra I – Fall 2007. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.1CEBE666

#### Recommended Additional Bibliography

- Linear algebra, Fraleigh, J. B., 1990