PřF:M2150 Algebra I - Course Information

M2150 Algebra I

Faculty of Science
Spring 2011

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)
doc. Mgr. Michal Kunc, Ph.D. (seminar tutor)
Mgr. Petr Liška, Ph.D. (assistant)

Guaranteed by

prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science

Timetable

Wed 11:00–12:50 A,01026

Timetable of Seminar Groups:

M2150/01: Tue 8:00–9:50 M4,01024, M. Kunc
M2150/02: Fri 10:00–11:50 M6,01011, O. Klíma
M2150/03: Fri 8:00–9:50 M4,01024, O. Klíma
M2150/04: Tue 16:00–17:50 M2,01021, 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 / plans the course is directly associated with

Mathematics (programme PřF, B-MA)

Course objectives

At the end of this course, students should be able to:
understand rudiments of group theory and ring theory;
explain basic notions and relations among them.

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 a 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 (normální podgrupa, faktorgrupa).
Centrum 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 a 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