Course: General and computer algebra

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Course title General and computer algebra
Course code KMA/XOPAL
Organizational form of instruction no contact
Level of course Doctoral
Year of study not specified
Semester Winter and summer
Number of ECTS credits 30
Language of instruction Czech
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Hora Jaroslav, doc. RNDr. CSc.
Course content
Solvability of algebraic equations, algebraically solvable equations. Outline of the Galois's theory - the impulse to foundation of general algebra. Examples of some algebraic structures - especially those with the finite coset. Boolean algebra and its models and applications. Finite groups. Groups of permutations, sign of permutation. Finite rings, fields. Field of quaternions. Factorization of polynomials, algorithmization (Kronecker's, Berlekamp's, Hensel's). Basics of computer integration. Solving systems of algebraic equations, problems of Mathematical Olympiad. Gröbner bases and Buchberger's algorithm.

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Monitoring
Learning outcomes
Solvability of algebraic equations, algebraically solvable equations. Outline of the Galois's theory - the impulse to foundation of general algebra. Examples of some algebraic structures - especially those with the finite coset. Boolean algebra and its models and applications. Finite groups. Groups of permutations, sign of permutation. Finite rings, fields. Field of quaternions. Factorization of polynomials, algorithmization (Kronecker's, Berlekamp's, Hensel's). Basics of computer integration. Solving systems of algebraic equations, problems of Mathematical Olympiad. Gröbner bases and Buchberger's algorithm.
Knowledge of basic notions of general and computer algebra.
Prerequisites
Knowledge of algebra in the extend of the basic university course.

Assessment methods and criteria
Oral examination, Written examination

Understanding of basic notions of the theory, knowledge of definitions, theorems and basic proofs. Ability to solve typical problems of a given theory. Written part of the exam is particularly oriented to verification whether a student can solve individual problems which come from a given theory. The aim of the oral part of the exam is to verify student's understanding of the theory and his/her ability to apply it.
Recommended literature
  • Geddes, K., O., Czapor, S., R., Labahn, G. Algorithms for Computer Algebra. Kluwer, 1992..
  • Procházka, L. a kol. Algebra. Praha: Academia, 1990..
  • von zur Gathen, G., Bernard, J. Modern Computer Algebra. Cambridge Univ. Press, 1999..
  • Weil, J. Rozpracovaná řešení úloh z vyšší algebry. Praha: Academia, 1987..


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Education Study plan (Version): Theory of Mathematics Education (2) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: -
Faculty: Faculty of Education Study plan (Version): Theory of Mathematics Education (2) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: -