MA0003 Algebra 1

Faculty of Education
Spring 2020
Extent and Intensity
2/2/0. 5 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Jaroslav Beránek, CSc. (lecturer)
RNDr. Břetislav Fajmon, Ph.D. (lecturer)
RNDr. Petra Antošová, Ph.D. (seminar tutor)
Mgr. Irena Budínová, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Jaroslav Beránek, CSc.
Department of Mathematics – Faculty of Education
Supplier department: Department of Mathematics – Faculty of Education
Timetable
Tue 14:00–15:50 učebna 1
  • Timetable of Seminar Groups:
MA0003/01: Mon 8:00–9:50 učebna 37, I. Budínová
MA0003/02: Thu 12:00–13:50 učebna 35, P. Antošová
MA0003/03: Thu 14:00–15:50 učebna 42, P. Antošová
Prerequisites
The subject is aimed at acquiring knowledge and skills in theory of binary algebraic operations, algebraic structures and their properties. THE PREREQUISITES ARE GOOD SKILLS IN THE SUBJECT "FOUNDATIONS OF MATHEMATICS" (MA0001).
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
Course objectives
At the end of the course the SS will be able to understand and explain the concepts of and solve problems in the following areas:
a) binary algebraic operations on a set, and their properties.
b) Algebraic structures with two operations, their substructures and properties.
c) Solution of algebraic equations.
d) Roots and powers of a complex number, calculation with complex numbers.
Learning outcomes
After the completion of the course the students will a) have knowledge of fundamental concepts in the theory of arithmetics, such as addition, product, intersection, union, operations with classes of decomposition of the set of all integers; b) have skills in solving algebraic equations in different areas of mathematics; c) know some methods of mathematical reasoning for binary operations and their properties; d) be acquainted with complex numbers, including the calculation of roots and powers of a complex number.
Syllabus
  • Syllabus:
  • Practice 1: Axioms of numerical operations, basic algebraic structures.
  • Practice 2: Determination of algebraic properties of a structure.
  • Lecture 1: Properties of groups, subgroups, group generators.
  • Lecture 2: Non-commutative groups.
  • Practice 3: Properties of groups, subgroups, group generators.
  • Lecture 3: Izomorfism, the Cayley theorem.
  • Practice 4: Non-commutative groups.
  • Lecture 4: The Lagrange theorem, group homomorfism.
  • Practice 5: Order of an element, cyclic groups. The congruence relation.
  • Lecture 5: Normal subgroup, factor group, operation on a factor group.
  • Practice 6: written test.
  • Lecture 6: Structurs with two operations: rings, integral domains, fields. .
  • Practice 7: Polynomials 01. Polynomial factorization, roots of a polynomial, the Horner scheme, greatest common divisor of polynomials.
  • Lecture 7: Structures with two operations: analogies of the Cayley theorem, the Lagrange theorem, fundemental theorem on homomorphisms, well-defined operation on a factor group. Extension of fields.
  • Practice 8: Polynomials 02. Rational roots of a polynomial. Finding.
  • Lecture 8: Polynomials -- algebraic metods of polynomial factorization.
  • Practice 9: Polynomials 03. Irrational and complex roots of a polynomial: metoda půlení intervalu, Newtonova metoda.
  • Lecture 9: Polynomy -- numerical methods of polynomial factorization. The R language.
  • Practice 10: Complex numbers 01: operations with complex numbers, algebraic and goniometric forms of a complex number.
  • Lecture 10: Vector space, its dimension and basis. Degrees of field extensions.
  • Practice 11: Complex numbers 02: n-th root and n-th power of a complex number, binomial equations.
  • Lecture 11: Construction and properties of sets N,Z.
  • Practice 12: Test b: polynomials, complex numbers.
  • Lecture 12: Construction and properties of sets Q,R,C.
Literature
    recommended literature
  • PINTER, Charles C. A book of abstract algebra. Second edition. Mineola, New York: Dover Publications, 2010, xiv, 384. ISBN 9780486474175. info
    not specified
  • HORÁK, Pavel. Cvičení z algebry a teoretické aritmetiky I. 2. vyd. Brno: Masarykova univerzita, 1998, 221 s. ISBN 8021018534. info
Teaching methods
Teaching methods chosen will reflect the contents of the subject and the level of students as newcomers to the university.
Assessment methods
The final mark comprises several parts all of which must be completed: a) practical part - two tests; b) the final written test
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
The textbooks used are: Pinter: The Book of Abstract Algebra. The teacher will provide the scan of some basic materials on polynomials and complex numbers.
The course is also listed under the following terms Spring 2018, Spring 2019, Spring 2021, Spring 2022, Spring 2023, Spring 2024, Spring 2025.
  • Enrolment Statistics (Spring 2020, recent)
  • Permalink: https://is.muni.cz/course/ped/spring2020/MA0003