Course details

Algebra

ALG FME BUT SOA Acad. year 2021/2022 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.

Guarantor

Deputy Guarantor

Language of instruction

Czech

Completion

Credit+Examination (written)

Time span

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

Assessment points

60 exam, 40 mid-term test

Department

Lecturer

Instructor

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.

Fundamental literature

  • 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)

Syllabus of lectures

  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

Syllabus of numerical exercises

  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

Syllabus of computer exercises

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

Progress assessment

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.


Controlled instruction

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.

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.

Exam prerequisites

The course-unit credit is awarded on condition of having attended the seminars actively and passed a written test.

Course inclusion in study plans

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