## M2150 Algebra I

Faculty of Science
Spring 2021
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)
Mgr. Pavel Francírek, Ph.D. (seminar tutor)
doc. Mgr. Ondřej Klíma, Ph.D. (seminar tutor)
doc. Mgr. Michal Kunc, Ph.D. (assistant)
Guaranteed by
Department of Mathematics and Statistics - Departments - Faculty of Science
Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science
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
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
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