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Lecturer(s)
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Trnečková Markéta, Mgr. Ph.D.
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Kühr Tomáš, Mgr. Ph.D.
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Kauer Martin, Mgr.
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Krupka Michal, doc. RNDr. Ph.D.
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Kolařík Miroslav, doc. RNDr. Ph.D.
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Lachman Dominik, Mgr.
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Course content
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1. Binary relations on sets. Groupoids, semigroups, monoids and groups. Homomorphisms and congruence, factorization. Subgroups and normal subgroups of groups. Congruence relations and homomorphisms of groups. Cyclic groups. Permutation groups, Cayley theorem. 2. Rings, integral domains and fields. Subrings and ideals, quotient ring. Homomorphism and congruence relations. Characteristic of a ring. Field of fractions of integral domains. 3. Divisibility in integral domains, basic properties. The existence of greatest common divisors. Euclidean and Gaussian integral domains. 4. Polynomials and polynomial functions over integral domains. Horner scheme. Properties of roots of polynomials. 5. Binomial equations. Algebraic equations and their algebraic solving. 6. Ordered sets. Sublattices, lattices, complete lattices. Modular and distributive lattices. Complemented lattices. Boolean algebras. Congruence relations and homomorphisms of lattices. 7. Application of relations, operations, polynomials and number fields in computer science. Application of lattice theory and group theory in computer science.
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Learning activities and teaching methods
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Lecture, Demonstration
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Learning outcomes
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The students become familiar with advanced concepts of algebra.
1. Knowledge Define basic notions and recall fundamental theorems of theory of groups and rings.
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Prerequisites
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KMI/ALG1 Algebra 1
KMI/ALG1
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Assessment methods and criteria
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Oral exam, Written exam
Active participation in class. Completion of assigned homeworks. Passing the oral (or written) exam.
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Recommended literature
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Halaš R., Chajda I. (1999). Cvičení z algebry. VUP Olomouc.
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Hort D., Rachůnek J. (2003). Algebra I. UP Olomouc.
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Chajda I. (1998). Algebra III. VUP Olomouc.
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Khanna V.K., Bhambri S.K. (2017). A Course in Abstract Algebra (fifth edition). Vikas Publishing.
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Krutský F. (1995). Algebra I.. VUP Olomouc.
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P., Emanovský, J. Kühr. (2007). Cvičení z algebry pro 1. ročník. UP Olomouc.
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Rachůnek, J. (2005). Grupy a okruhy. VUP Olomouc.
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