| Course title | Algebra II |
|---|---|
| Course code | KAG/KALII |
| Organizational form of instruction | Seminar |
| Level of course | Master |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 7 |
| Language of instruction | Czech |
| Status of course | unspecified |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
|---|
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| Course content |
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1. Binary relations, reflexive, symmetric and transitive relations. Equivalence relations and partitions, quotient sets. 2. Grupoids, semigroups and groups. Natural and integer powers in semigroups and groups. Homomorphisms and congruence relations, quotient grupoids, the homomorphism theorem for grupoids. Subgroups and normal subgroups of groups. Congruence relations and homomorphisms of groups. Quotient groups. The homomorphism theorem for groups, isomorphism theorems. Subgroup generated by a set, order of a subgroup and of an element. Cyclic groups. Permutation groups, the Cayley theorem. Direct products of groupoids. 3. Rings, integral domains and fields. Ideals, prime ideals and maximal ideals. Homomorphism and congruence relations, quotient rings. The homomorphism theorem. Order of an element, characteristic of a ring. 5. Divisibility in integral domains. Units, irreducible and prime elements. Greatest common divisor, least common multiple. Ideal generated by a set, pricipal ideal domains. Euclidean domains, Gaussian domains. 6. Partially ordered sets. Mappings of partially ordered sets: monotone, antitone, isomorphic embedding, isomorphism. Distinguished elements: maximal, minimal, greatest, least. Lower and upper cone of a set, directed sets. Supremum and infimum, semilattices. The Zorn lemma. 7. Lattices: Partially ordered sets and algebras. Complete lattices, the fixed point theorem. Sublattices. Lattice homomorphisms and congruence relations. Quotient lattices, homomorphism theorem. Ideals (and filters) of lattices. Ideal generated by a set, principal ideals.
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| Learning activities and teaching methods |
| Lecture, Monologic Lecture(Interpretation, Training) |
| Learning outcomes |
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Understand basic elements of theory of groups and rings. Understand divisibility theory in integral domains and basics of lattice theory.
1. Knowledge Define basic notions, describe basic constructions and recall fundamental theorems of theory of groups and rings. Comprehension of basics of divisibility theory in integral domains and basics of lattice theory. |
| Prerequisites |
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unspecified
KAG/KALI |
| Assessment methods and criteria |
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Oral exam, Written exam
Exam: the student has to understand the subject and prove basic statements |
| Recommended literature |
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| Study plans that include the course |