Course detail

General Algebra

FSI-SOACompulsoryBachelor's (1st cycle)Acad. year: 2016/2017Summer semester1. year of study5  credits

The course will familiarise students with basics of modern algebra. We will describe general properties of universal algebras and study, in more detail, individual algebraic structures, i.e., groupoids, semigroups, monoids, groups, rings and fields. Particular emphasis will be placerd on groups, rings (especially the ring of polynomials) and finite (Galois) fields.

Learning outcomes of the course unit

Students will be made familiar with the basics of general algebra. It will help them to realize numerous mathematical connections and therefore to understand different mathematical branches. The course will provide students also with useful tools for various applications.

Mode of delivery

90 % face-to-face, 10 % distance learning

Prerequisites

The students are supposed to be acquainted with the fundamentals of linear algebra taught in the first semester of the bachelor's study programme.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

L.Procházka a kol.: Algebra, Academia, Praha, 1990
S.Lang, Undergraduate Algebra, Springer-Verlag,1990
G.Gratzer: Universal Algebra, Princeton, 1968
A.G.Kuroš, Kapitoly z obecné algebry, Academia, Praha, 1977
S. MacLane a G. Birkhoff, Algebra, Vyd. tech. a ekon. lit., Bratislava, 1973 (CS)
S.MacLane, G.Birkhoff: Algebra, Alfa, Bratislava, 1973
J. Karásek and L. Skula, Obecná algebra (skriptum), Akademické nakladatelství CERM, 2008 (CS)
S. Lang, Undergraduate Algebra (2nd Ed.), Springer-Verlag, New York-Berlin-Heidelberg, 1990 (EN)
J.Šlapal, Základy obecné algebry, Ústav matematiky FSI VUT v Brně, 2013 - elektronický text (CS)
Procházka a kol., Algebra, Academia, Praha, 1990

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures an on getting acquainted with algebraic software.

Assesment methods and criteria linked to learning outcomes

The course-unit credit is awarded on condition of having attended the seminars actively and passed a written test. The exam has a written and an oral part. The written part tests student's ability to deal with various problems using the knowledge and skills acquired in the course. In the oral part, the student has to prove that he or she has mastered the related theory.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The aim of the course is to provide students with the fundamentals of modern algebra, i.e., with the usual algebraic structures and their properties. These structures often occur in various applications and it is therefore necessary for the students to have a good knowledge of them.

Specification of controlled education, way of implementation and compensation for absences

Since the attendance at seminars is required, it will be checked systematically by the teacher supervising the seminar. If a student misses a seminar, an excused absence can be compensated for via make-up topics of exercises.

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Operations and laws, the concept of a universal algebra
2. Some important types of algebras
3. Basics of the group theory
4. Subalgebras, decomposition of a group (by a subgroup)
5. Homomorphisms and isomorphisms
6. Congruences and quotient algebras
7. Congruences on groups and rings
8. Direct products of algebras
9. Ring of polynomials
10.Divisibility and integral domains
11.Gaussian and Euclidean rings
12.Mimimal fields, field extensions
13.Galois fields

seminars

22 hours, compulsory

Teacher / Lecturer

Syllabus

1. Operations, algebras and types
2. Basics of the groupoid and group theories
3. Subalgebras, direct products and homomorphisms
4. Congruences and factoralgebras
5. Congruence on groups and rings
6. Rings of power series and of polynomials
7. Polynomials as functions, interpolation
8. Divisibility and integral domains
9. Gauss and Euclidean Fields
10. Minimal fields, field extensions
11. Construction of finite fields

seminars in computer labs

4 hours, compulsory

Teacher / Lecturer

Syllabus

1. Using software Maple for solving problems of general algebry
2. Using software Mathematica for solving problems of general algebra