Groups and Rings (FSI-SG0)

Academic year 2018/2019
Supervisor: doc. RNDr. Miroslav Kureš, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
Students will be made familiar with advanced algebra, in particular group theory and ring theory.
Learning outcomes and competences:
The course makes access to mastering in a wide range of results of algebra.
Prerequisites:
Linear algebra, general algebra
Course contents:
In the course Groups and rings, students are familiarised with selected topics of algebra. The acquired knowledge is a starting point not only for further study of algebra and other mathematical disciplines, but also a necessary assumption for a use of algebraic methods in a practical solving of number of problems.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline.
Assesment methods and criteria linked to learning outcomes:
Course credit: the attendance, satisfactory solutions of homeworks
Controlled participation in lessons:
Lectures: recommended
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
Course curriculum:
    Lecture 1. Groups, subgroups, factor groups
2. Group homomorphisms, group actions on a set, group products
3. Topological, Lie and algebraic groups
4. Jets of mappings, jet groups
5. Rings and ideals
6. Euclidean rings, PID and UFD
7. Monoid a group rings
8. Gradede rings, R-algebras
9. Polynomials and polynomial morphisms
10. Modules and representations
11. Finite group and rings
12. Quaternionic algebras
13. Reserve - the topic to be specified
Literature - fundamental:
1. M.F. Atiyah and I.G. Macdonald, Introduction To Commutative Algebra, Addison-Wesley series in mathematics, Verlag Sarat Book House, 1996
2. O. Bogopolski, Introduction to Group Theory, European Mathematical Society 2008
3. G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, Springer 2002
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3A-P full-time study B-MAI Mathematical Engineering -- Cr 0 Elective (voluntary) 1 2 W
B3A-P full-time study B-FIN Physical Engineering and Nanotechnology -- Cr 0 Elective (voluntary) 1 2 W