Course detail

General Algebra

FSI-SOAAcad. year: 2019/2020

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), integral domains 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.

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 (EN)
G.Gratzer: Universal Algebra, Princeton, 1968 (EN)
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 (EN)
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 (CS)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the general algebrta. Exercises are focused on practical understanding of the topics presented in lectures by means of examples and also 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.

Classification of course in study plans

  • Programme B-MAI-P Bachelor's, 1. year of study, summer semester, 5 credits, compulsory

  • Programme MITAI Master's

    specialization NADE , any year of study, summer semester, 5 credits, optional
    specialization NBIO , any year of study, summer semester, 5 credits, optional
    specialization NGRI , any year of study, summer semester, 5 credits, optional
    specialization NNET , any year of study, summer semester, 5 credits, optional
    specialization NVIZ , any year of study, summer semester, 5 credits, optional
    specialization NCPS , any year of study, summer semester, 5 credits, optional
    specialization NSEC , any year of study, summer semester, 5 credits, optional
    specialization NEMB , any year of study, summer semester, 5 credits, optional
    specialization NHPC , any year of study, summer semester, 5 credits, optional
    specialization NISD , any year of study, summer semester, 5 credits, optional
    specialization NIDE , any year of study, summer semester, 5 credits, optional
    specialization NISY , any year of study, summer semester, 5 credits, optional
    specialization NMAL , any year of study, summer semester, 5 credits, optional
    specialization NMAT , any year of study, summer semester, 5 credits, compulsory
    specialization NSEN , any year of study, summer semester, 5 credits, optional
    specialization NVER , any year of study, summer semester, 5 credits, optional
    specialization NSPE , any year of study, summer semester, 5 credits, optional

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, basics of the group theory
3. Subalgebras, decomposition of a group (by a subgroup)
4. Homomorphisms and isomorphisms
5. Congruences and quotient algebras
6. Congruences on groups and rings
7. Direct products of algebras
8. Ring of polynomials
9.Integral domains and divisibility, Gauss rings
10. Rings of principal ideals, Euclidean rings
11.Divisibility fields of integral domains, minimal fields
12.Root fields and field extensions
13.Decomposition and Galois fields

Exercise

22 hours, compulsory

Teacher / Lecturer

Syllabus

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

Computer-assisted exercise

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

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