M2150 Algebra I

Faculty of Science
Spring 2023
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, Ph.D. (assistant)
Mgr. Jan Vondruška (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
Tue 10:00–11:50 M1,01017
  • Timetable of Seminar Groups:
M2150/01: Thu 14:00–15:50 M1,01017, O. Klíma
M2150/02: Thu 8:00–9:50 M1,01017, O. Klíma
M2150/03: Fri 8:00–9:50 M6,01011, O. Klíma
Prerequisites (in Czech)
! MUC32 Algebra && !(NOW( MUC32 Algebra ))
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
Further Comments
Study Materials
The course can also be completed outside the examination period.
The course is taught annually.
Listed among pre-requisites of other courses
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Autumn 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2023, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2023/M2150