Title:  Algebra 

Code:  ALG (FSI SOA) 

Ac.Year:  2019/2020 

Sem:  Summer 

Curriculums:  

Language of Instruction:  Czech 

Credits:  5 

Completion:  credit+exam (written) 

Type of instruction:  Hour/sem  Lectures  Seminar Exercises  Laboratory Exercises  Computer Exercises  Other 

Hours:  26  22  0  4  0 

 Exams  Tests  Exercises  Laboratories  Other 

Points:  60  40  0  0  0 



Guarantor:  Šlapal Josef, prof. RNDr., CSc. (DADM) 

Deputy guarantor:  Rogalewicz Adam, doc. Mgr., Ph.D. (DITS) 

Lecturer:  Šlapal Josef, prof. RNDr., CSc. (DADM) 
Instructor:  Šlapal Josef, prof. RNDr., CSc. (DADM) 

Faculty:  Faculty of Mechanical Engineering BUT 

Department:  Department of Algebra and Discrete Mathematics FME BUT 

 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.  Description: 

  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.  Knowledge and skills required for the course: 

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

 
Operations and laws, the concept of a universal algebra

Some important types of algebras

Basics of the group theory

Subalgebras, decomposition of a group (by a subgroup)

Homomorphisms and isomorphisms

Congruences and quotient algebras

Congruences on groups and rings
 Direct products of algebras

Ring of polynomials

Divisibility and integral domains

Gaussian and Euclidean rings

Mimimal fields, field extensions

Galois fields
 Syllabus of numerical exercises: 

  Operations, algebras and types

Basics of the groupoid and group theories

Subalgebras, direct products and homomorphisms

Congruences and factoralgebras

Congruence on groups and rings

Rings of power series and of polynomials

Polynomials as functions, interpolation

Divisibility and integral domains

Gauss and Euclidean Fields

Minimal fields, field extensions

Construction of finite fields
 Syllabus of computer exercises: 

 
Using software Maple for solving problems of general algebry

Using software Mathematica for solving problems of general algebra
 Fundamental literature: 

  S.Lang, Undergraduate Algebra, SpringerVerlag,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
 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.), SpringerVerlag, New YorkBerlinHeidelberg, 1990
 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 makeup
topics of exercises.  Progress assessment: 

  The courseunit 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.  
