General Algebra (FSI-SOA)

Academic year 2019/2020
Supervisor: prof. RNDr. Josef Šlapal, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Course type: departmental course, applied basis
Aims of the course unit:
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.
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.
Prerequisites:
The students are supposed to be acquainted with the fundamentals of linear algebra taught in the first semester of the bachelor's study programme.
Course contents:
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), integral domains and finite (Galois) fields.
Teaching methods and criteria:
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.
Assesment methods and criteria linked to learning outcomes:
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 participation in lessons:
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.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  11 × 2 hrs. compulsory                  
    Computer-assisted exercise  2 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 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
    Exercise 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
    Computer-assisted exercise 1. Using software Maple for solving problems of general algebry
2. Using software Mathematica for solving problems of general algebra
Literature - fundamental:
1. S.Lang, Undergraduate Algebra, Springer-Verlag,1990
2. G.Gratzer: Universal Algebra, Princeton, 1968
3. S.MacLane, G.Birkhoff: Algebra, Alfa, Bratislava, 1973
4. J. Karásek and L. Skula, Obecná algebra (skriptum), Akademické nakladatelství CERM, Brno 2008
5. J.Šlapal, Základy obecné algebry (skriptum), Akademické nakladatelství CERM, Brno 2022.
6. Procházka a kol., Algebra, Academia, Praha, 1990
Literature - recommended:
1. L.Procházka a kol.: Algebra, Academia, Praha, 1990
2. A.G.Kuroš, Kapitoly z obecné algebry, Academia, Praha, 1977
3. 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
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
MITAI full-time study NBIO Bioinformatics and Biocomputing -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NISY Intelligent Systems -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NSEN Software Engineering -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NVIZ Computer Vision -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NGRI Computer Graphics and Interaction -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NISD Information Systems and Databases -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NSEC Cybersecurity -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NCPS Cyberphysical Systems -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NHPC High Performance Computing -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NNET Computer Networks -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NMAL Machine Learning -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NVER Software Verification and Testing -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NIDE Intelligent Devices -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NEMB Embedded Systems -- Cr,Ex 5 Elective 1 0 S
B-MAI-P full-time study --- no specialisation -- Cr,Ex 5 Compulsory 1 1 S
MITAI full-time study NSPE Sound, Speech and Natural Language Processing -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NADE Application Development -- Cr,Ex 5 Elective 1 0 S
MITAI full-time study NMAT Mathematical Methods -- Cr,Ex 5 Compulsory 1 0 S