Course details
General Algebra
SOA FSI SOA Acad. year 2024/2025 Summer semester 5 credits
The course will familiarise students with basics of modern algebra. We will describe general properties and fonstructions of universal algebras like subalgebras, homomorphisms, and factorization. In more detail, individual algebraic structures will be studied such as groupoids, semigroups, monoids, groups, rings and fields. Particular emphasis will be placerd on groups, rings (especially the ring of polynomials), integral domains (including divisibility) and finite (Galois) fields.
Guarantor
Language of instruction
Completion
Time span
- 26 hrs lectures
- 26 hrs exercises
Department
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.
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.
Prerequisite knowledge and skills
The students are supposed to be acquainted with the fundamentals of linear algebra taught in the first semester of the bachelor's study programme.
Study literature
- 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
Fundamental literature
- S.Lang, Undergraduate Algebra, Springer-Verlag,1990
- G.Gratzer: Universal Algebra, Princeton, 1968
- S.MacLane, G.Birkhoff: Algebra, Alfa, Bratislava, 1973
Syllabus of lectures
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
Syllabus of exercises
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
Syllabus of numerical exercises
1. Using software Maple for solving problems of general algebry
2. Using software Mathematica for solving problems of general algebra
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.
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.
How to contact the teacher
During teaching the students are in personal contact with the teacher.
Course inclusion in study plans