Course detail
General Algebra
FSI-SOAAcad. year: 2025/2026
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
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
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
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
Prerequisites and corequisites
Basic literature
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
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
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
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
Teacher / Lecturer
Syllabus
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
Teacher / Lecturer
Syllabus
2. Using software Mathematica for solving problems of general algebra