Title:

# Algebra

Code:ALG (FSI SOA)
Ac.Year:2019/2020
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
MITAINBIO-Elective
MITAINCPS-Elective
MITAINEMB-Elective
MITAINGRI-Elective
MITAINHPC-Elective
MITAINIDE-Elective
MITAINISD-Elective
MITAINISY-Elective
MITAINMAL-Elective
MITAINMAT-Compulsory
MITAINNET-Elective
MITAINSEC-Elective
MITAINSEN-Elective
MITAINSPE-Elective
MITAINVER-Elective
MITAINVIZ-Elective
Language of Instruction:Czech
Credits:5
Completion:credit+exam (written)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:2622040
ExamsTestsExercisesLaboratoriesOther
Points:6040000
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:

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
Syllabus of computer exercises:

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

• 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.), Springer-Verlag, New York-Berlin-Heidelberg, 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 make-up topics of exercises.
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.