## MUC32 Algebra

**Faculty of Science**

Spring 2022

**Extent and Intensity**- 2/2/0. 5 credit(s). Type of Completion: zk (examination).

Taught in person. **Teacher(s)**- prof. RNDr. Radan Kučera, DSc. (lecturer)

doc. Mgr. Ondřej Klíma, Ph.D. (seminar tutor)

Mgr. Pavel Francírek, Ph.D. (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 **Prerequisites**(in Czech)- !
**M2150**Algebra I && !( NOW (**M2150**Algebra I )) **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**- there are 9 fields of study the course is directly associated with, display
**Course objectives**- The aim of this introductory course is to give students the basic algebraic rudiments, which are assumed in some advanced courses.
**Learning outcomes**- 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**- Divisibility of integers, Euclidean algorithm, Bezout identity.
- Congruence, residue class, addition and multiplication.
- Binary operation on a set, semigroup, (abelian) group; examples of groups and semigroups (numbers, permutations, residue classes, matrices, vectors).
- (Commutative) ring, integral domain, field, their basic properties.
- Basic properties of groups (including the notions of a power and the order of an element, the exponent of a group).
- Subgroup (including the subgroup generated by a set), cyclic group.
- Homomorphism and isomorphism of groups, the product of groups, the classification of cyclic groups and of finite abelian groups).
- Partition of a group, left cosets of a subgroup (Lagrange's theorem and its consequences).
- Subring (including the subring generated by a set).
- Homomorphism and isomorphism of rings, the product of rings.
- Divisibility in commutative rings.
- Polynomials (basic properties, division of polynomials with remainder, Euclidean algorithm, the 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).
- Polynomials in several variables, symetrical polynomials, elementary symetrical polynomials and their application.

**Literature**- ROSICKÝ, Jiří.
*Algebra*. 4., přeprac. vyd. Brno: Masarykova univerzita, 2002. 133 s. ISBN 80-210-2964-1. info

- ROSICKÝ, Jiří.
**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****Further Comments**- The course can also be completed outside the examination period.

The course is taught annually.

The course is taught: every week. **Listed among pre-requisites of other courses****M2150**Algebra I

!MUC32 && !(NOW(MUC32))**M3150**Algebra II

M2150 || MUC32

**Teacher's information**- http://math.muni.cz/~klima/Algebra/algI-prf-jaro15.html

- Enrolment Statistics (Spring 2022, recent)
- Permalink: https://is.muni.cz/course/sci/spring2022/MUC32