General Algebra

• 26 hrs lectures
• 26 hrs exercises

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.

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.

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

• L.Procházka a kol.: Algebra, Academia, Praha, 1990
• A.G.Kuroš, Kapitoly z obecné algebry, Academia, Praha, 1977
• S. Lang, Undergraduate Algebra (2nd Ed.), Springer-Verlag, New York-Berlin-Heidelberg, 1990

• S.Lang, Undergraduate Algebra, Springer-Verlag,1990
• G.Gratzer: Universal Algebra, Princeton, 1968
• S.MacLane, G.Birkhoff: Algebra, Alfa, Bratislava, 1973

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

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

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

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.

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.

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.