Course detail

General Algebra

FSI-SOAAcad. year: 2024/2025

The course will familiarise students with basics of modern algebra. We will describe general properties and fonstructions of universal algebras like subalgebras, homomorphisms, and factorization. In more detail, individual algebraic structures will be studied such as  groupoids, semigroups, monoids, groups, rings and fields. Particular emphasis will be placerd on groups, rings (especially the ring of polynomials), integral domains (including divisibility) and finite (Galois) fields.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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

Rules for evaluation and completion of the course

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.


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.

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.

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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

G.Gratzer: Universal Algebra, Princeton, 1968 (EN)
J. Karásek and L. Skula, Obecná algebra (skriptum), Akademické nakladatelství CERM, Brno 2008 (CS)
J.Šlapal, Základy obecné algebry (skriptum), Akademické nakladatelství CERM, Brno 2022. (CS)
Procházka a kol., Algebra, Academia, Praha, 1990 (CS)
S.Lang, Undergraduate Algebra, Springer-Verlag,1990 (EN)
S.MacLane, G.Birkhoff: Algebra, Alfa, Bratislava, 1973 (EN)

Recommended reading

A.G.Kuroš, Kapitoly z obecné algebry, Academia, Praha, 1977
L.Procházka a kol.: Algebra, Academia, Praha, 1990
S. Lang, Undergraduate Algebra (2nd Ed.), Springer-Verlag, New York-Berlin-Heidelberg, 1990 (EN)
S. MacLane a G. Birkhoff, Algebra, Vyd. tech. a ekon. lit., Bratislava, 1973 (CS)

Classification of course in study plans

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

  • Programme MITAI Master's

    specialization NMAT , 0 year of study, summer semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CLS , 1 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

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 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

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 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

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