MA0003 Algebra 1

Faculty of Education
Spring 2024
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)
Guaranteed by
RNDr. Břetislav Fajmon, Ph.D.
Department of Mathematics – Faculty of Education
Supplier department: Department of Mathematics – Faculty of Education
Timetable
Mon 14:00–15:50 učebna 30
  • Timetable of Seminar Groups:
MA0003/PrezSem01: Tue 15:00–16:50 učebna 72, P. Antošová
MA0003/PrezSem02: Thu 13:00–14:50 učebna 32, P. Antošová
MA0003/PrezSem03: Thu 15:00–16:50 učebna 32, P. Antošová
MA0003/T01: Wed 21. 2. to Fri 31. 5. Wed 11:00–12:50 118, P. Antošová, Nepřihlašuje se. Určeno pro studenty se zdravotním postižením.
Prerequisites
The subject is aimed at acquiring knowledge and skills in theory of binary algebraic operations, algebraic structures and their morphisms. Getting acquainted with the theory of cyclic groups and factoring structures forms an integral part. 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: binary algebraic operations on a set, and their properties. Algebraic structures with one operation, their substructures and homomorphisms. Algebraic structures with two operations, their substructures and homomorphisms. Cyclic groups. Factoring structures (generating partition, normal subgroup, quotient group, left and right cosets for a subgroup, cosets for an ideal, quotient ring. Last but not least, the acquire ability to find roots of a polynomial, calculate roots and powers of 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:
  • Lecture 1: Binary operations - examples.
  • Practice 1: Axioms of binary operations with numbers. Basic algebraic structures.
  • Lecture 2: Properties of groups, subgroups, generating a subgroup.
  • Practice 2: Properties of binary operations.
  • Lecture 3: Noncommutative groups.
  • Practice 3: Properties of groups, subgroups, group generators. Order of an element, cyclic groups, congruence relation.
  • Lecture 4: Izomorfisms, Cayley theorem.
  • Practice 4: Noncommutative groups.
  • Lecture 5: Lagrange theorem, group homomorfisms.
  • Practice 5: Algebraic structures with two operations.
  • Lecture 6: Struktures with two operations: rings, integral domains, fields.
  • Practice 6: Polynomials 01. Decomposition of a polynomial into a product of its factors, Horner scheme, sign changes, rational roots of polynomials.
  • Lecture 7: Polynomials -- summary of algebraic methods.
  • Practice 7: Polynomials 02. Greatest common divisor, Eucleidian algorithm. Multiple roots of a polynomial.
  • Lecture 8: Polynomials -- summary of numerical methods for finding roots of a polynomial.
  • Practice 8: Polynomials 03. Numerical methods: bisection, Newton method.
  • Lecture 9: Peano set and its axioms.
  • Practice 9: A test.
  • Lecture 10: Algebraic structures with sets of numbers.
  • Practice 10: Complex numbers - introduction.
  • Lecture 11: Review.
  • Practice 11: Complex numbers 02: n-th power and n-th root of a complex number, binomial equations.
  • Lecture 12: Review 02.
  • Cvičení 12: Complex numbers 03: equations with complex coefficients.
Literature
    recommended literature
  • PINTER, Charles C. A book of abstract algebra. Second edition. Mineola, New York: Dover Publications, 2010, xiv, 384. ISBN 9780486474175. info
  • 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 exam comprises several parts all of which must be completed: a) practical part - one test during the 9th lesson; students must complete at least 60 per cent of the tasks; maximum number of points is 30. b) final written test - muximum number of points is 70; students must complete at least 50 per cent of the tasks. c) the mark will be created as the sum of the preceding two numbers of points provided the minimum levels of both required parts are reached or superceded.
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
The course is also listed under the following terms Spring 2018, Spring 2019, Spring 2020, Spring 2021, Spring 2022, Spring 2023, Spring 2025.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/ped/spring2024/MA0003