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..
