General Algebra (FSI-SOA-A)

Academic year 2018/2019
Supervisor: prof. RNDr. Josef Šlapal, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
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) and finite (Galois) fields.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures an 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
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
    Exercise 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
    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
Literature - recommended:
1. L.Procházka a kol.: Algebra, Academia, Praha, 1990
2. A.G.Kuroš, Kapitoly z obecné algebry, Academia, Praha, 1977
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3A-A full-time study B-MAI Mathematical Engineering -- Cr,Ex 5 Compulsory 1 1 S