## M2150 Algebra I

Faculty of Science
Spring 2017
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer)
doc. Mgr. Ondřej Klíma, Ph.D. (seminar tutor)
Mgr. Pavel Francírek (assistant)
Guaranteed by
prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics - Departments - Faculty of Science
Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science
Timetable
Mon 20. 2. to Mon 22. 5. Thu 8:00–9:50 M1,01017
• Timetable of Seminar Groups:
M2150/01: Mon 20. 2. to Mon 22. 5. Tue 12:00–13:50 M5,01013, O. Klíma
M2150/02: Mon 20. 2. to Mon 22. 5. Fri 8:00–9:50 M6,01011, O. Klíma
M2150/03: Mon 20. 2. to Mon 22. 5. Tue 16:00–17:50 M5,01013, O. Klíma
Prerequisites (in Czech)
! M2155 Algebra 1
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
Fields of study the course is directly associated with
there are 9 fields of study the course is directly associated with, display
Course objectives
At the end of this course, students should be able to:
* define basic notions of group theory and ring theory;
* explain learned theoretical results;
* apply learned methods to concrete exercises.
Syllabus
• Binary operation on a set, semigroup, (abelian) group; examples of groups and semigroups (numbers, permutations, residue classes, matrices, vectors), basic properties of groups (including powers and order of an element).
• Subgroup (including the subgroup generated by a set).
• Homomorphism and isomorphism of groups (Cayley's theorem, classification of cyclic groups), product of groups.
• Partition of a group, left cosets of a subgroup (Lagrange's theorem and their consequences).
• Quotient groups (normal subgroup, quotient group).
• Center of a group.
• Finite groups, p-groups, classification of finite abelian groups, Sylow's theorems.
• (Commutative) ring, integral domain, fields, their basic properties.
• Subring (including the subring generated by a set).
• Homomorphism and isomorphism of rings.
• Polynomials (basic properties, division of polynomials with remainder, Euclidean algorithm, value of a polynomial in an element, root of a polynomial, multiple roots, connection with the derivative of a polynomial).
• Polynomials over the fields of complex, real and rational numbers and over the ring of integers (irreducible polynomials, computation of roots of a polynomial).
Literature
• ROSICKÝ, Jiří. Algebra. 4., přeprac. vyd. Brno: Masarykova univerzita, 2002. 133 s. ISBN 80-210-2964-1. info
Teaching methods
Lectures: theoretical explanation. Exercises: solving problems with the aim to understand basic concepts and theorems, homework (e-tests).
Assessment methods
Examination consists of two parts: a written test and an oral examination. To pass the written part, which consists of 7 exercises, it is necessary to get at least 50% of points (35 points of 70). The students successful in the written part have to show in the following oral part that they are able to define the used notions and to work with them, to formulate the explained statements and to prove the easier of them.
Language of instruction
Czech
Follow-Up Courses