M2150 Algebra I

Faculty of Science
Spring 2015
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)
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
Fri 8:00–9:50 M1,01017
  • Timetable of Seminar Groups:
M2150/01: Wed 14:00–15:50 M5,01013, O. Klíma
M2150/02: Fri 10:00–11:50 M1,01017, O. Klíma
M2150/03: Tue 10:00–11:50 M1,01017, O. Klíma
Prerequisites (in Czech)
! M2155 Algebra 1
Course Enrolment Limitations
The course is offered to students of any study field.
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
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
Teacher's information
http://math.muni.cz/~klima/Algebra/algI-prf-jaro15.html
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 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2015, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2015/M2150