Course detail
General Algebra
FSI-SOA Acad. year: 2021/2022 Summer semester
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.
Language of instruction
Czech
Number of ECTS credits
5
Supervisor
Department
Learning outcomes of the course unit
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.
Planned learning activities and teaching methods
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.
Aims
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.
Specification of controlled education, way of implementation and compensation for absences
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.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's, compulsory
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NADE: Application Development, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NBIO: Bioinformatics and Biocomputing, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NCPS: Cyberphysical Systems, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NEMB: Embedded Systems, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NGRI: Computer Graphics and Interaction, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NHPC: High Performance Computing, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NIDE: Intelligent Devices, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NISD: Information Systems and Databases, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NISY: Intelligent Systems, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NMAL: Machine Learning, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NMAT: Mathematical Methods, compulsory
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NNET: Computer Networks, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NSEC: Cybersecurity, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NSEN: Software Engineering, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NSPE: Sound, Speech and Natural Language Processing, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NVER: Software Verification and Testing, elective
Programme MITAI: Information Technology and Artificial Intelligence, Master's
specialization NVIZ: Computer Vision, elective
Type of course unit
Lecture
26 hours, optionally
Teacher / Lecturer
Syllabus
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
22 hours, compulsory
Teacher / Lecturer
Syllabus
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
4 hours, compulsory
Syllabus
1. Using software Maple for solving problems of general algebry
2. Using software Mathematica for solving problems of general algebra