Faculty of Information Technology, BUT

Course details


ALG FME BUT SOA Acad. year 2019/2020 Summer semester 5 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.


Deputy Guarantor

Language of instruction



Credit+Examination (written)

Time span

26 hrs lectures, 22 hrs exercises, 4 hrs pc labs

Assessment points

60 exam, 40 half-term test





Generic learning outcomes and competences

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.

Learning objectives

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.

Prerequisite kwnowledge and skills

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

Study literature

  • L.Procházka a kol.: Algebra, Academia, Praha, 1990
  • A.G.Kuroš, Kapitoly z obecné algebry, Academia, Praha, 1977
  • S. MacLane a G. Birkhoff, Algebra, Vyd. tech. a ekon. lit., Bratislava, 1973 4. S. Lang, Undergraduate Algebra (2nd Ed.), Springer-Verlag, New York-Berlin-Heidelberg, 1990

Fundamental literature

  • S.Lang, Undergraduate Algebra, Springer-Verlag,1990
  • G.Gratzer: Universal Algebra, Princeton, 1968
  • S.MacLane, G.Birkhoff: Algebra, Alfa, Bratislava, 1973
  • J. Karásek and L. Skula, Obecná algebra (skriptum), Akademické nakladatelství CERM, 2008
  • J.Šlapal, Základy obecné algebry, Ústav matematiky FSI VUT v Brně, 2013 - elektronický text
  • Procházka a kol., Algebra, Academia, Praha, 1990

Syllabus of lectures

  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

Syllabus of numerical exercises

  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

Progress assessment

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.

Controlled instruction

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.

Course inclusion in study plans

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