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Lecturer(s)
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Zalabová Lenka, doc. Mgr. Ph.D.
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Course content
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We study algebraic structures with one and two binary operations. We give basic definitions and properties and we study important cases of these structures. Summary: Algebraic structures with one operation - grupoids, semigropus, groups. Subgroups, homomorphisms, isomorphisms. Group of permutations. Symmetry groups. Cyclic and dihedral groups. Congruence relation and groups of congruence classes. Divisibility of integers. Euklid agorithm. Bezout equality and consequences. Primes. Linear congruence equations. Euller theorem. Applications in crypthography. Algebraic structures with two operations - rings, domains, fields. Rings/fields of numbers. Rings/fields of congruence classes. Polynomials, roots of polynomials, irreducibility. Complex polynomias, fundamental theorem of algebra. Rational and real polynomials. Eisenstein criterion.
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Learning activities and teaching methods
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Monologic (reading, lecture, briefing), Dialogic (discussion, interview, brainstorming), Work with text (with textbook, with book), Individual preparation for exam
- Class attendance
- 56 hours per semester
- Preparation for classes
- 56 hours per semester
- Preparation for exam
- 56 hours per semester
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Learning outcomes
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The goal of the course is the introduction into elementary algebraic structures.
The student will acquire the basic knowledge of abstract algebra and number theory.
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Prerequisites
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The knowledge of mathematics on the level of secondary school and the theory of matrices and their properties.
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Assessment methods and criteria
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Combined exam, Seminar work
Active participation in the tutorials, a project focused on mathematic olympiad or applied algebra (including programming), passing both written and oral part of the exam (50%).
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Recommended literature
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J. Blažek, E. Calda, M. Koman, B. Kussová, Algebra a teoretická aritmetika 1, SPN Praha, 1983..
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J. Blažek, M. Koman, B. Vojtášková, Algebra a teoretická aritmetika 2, SPN Praha, 1985..
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J. Procházka a kolektiv, Polynomy, Jihočeská univerzita, České Budějovice, 1985..
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J. Rosický: Algebra, Masarykova univerzita, Brno, 2007..
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P. Tlustý: Obecná algebra pro učitele, Jihočeská univerzita, České Budějovice, 2006..
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Skripta ke kurzu UMB567 Algebra.
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Lenka Zalabová. Skripta ke kurzu UMB567 Algebra.
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