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Lecturer(s)
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Zdráhal Tomáš, doc. RNDr. CSc.
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Course content
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Rings, definitions and fundamental properties, matrix rings, subrings. Polynomial rings, transcendental elements, degree of polynomials. Integral domains and fields, zero divisors, field characteristic. The quotient field of an integral domain, construction of fractions and the field of rational numbers. Ideals and factor rings, congruences on rings, prime and maximal ideals. Solving algebraic equations, equivalent and consequential transformations.
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Learning activities and teaching methods
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Lecture
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Learning outcomes
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The objective is to introduce students to fundamental and advanced concepts of general algebra, focusing on the theory of rings, integral domains, and fields. The course builds upon group theory and extends it to structures with two binary operations. Emphasis is placed on understanding the factorization of structures, the construction of number domains (e.g., quotient fields), and the application of algebraic knowledge in solving equations.
Upon completion, the student will be able to: Define and identify basic types of rings, integral domains, and fields. Perform operations in rings of polynomials and matrices. Construct quotient fields from integral domains. Work with ideals and construct factor rings. Distinguish between equivalent and non-equivalent (consequential) transformations when solving algebraic equations.
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Prerequisites
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Successful completion of the course Algebra 1.
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Assessment methods and criteria
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Mark
Active knowledge of the course subject matter. Specific requirements (including individual assignments for term papers) are specified in the respective MS Teams group. Regarding active physical attendance, a minimum of 49% is required for both full-time and part-time students. For students with an Individual Study Plan (ISP), the minimum attendance is 20% (specific ISP requirements must be consulted in person with the instructor at the beginning of the semester).
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Recommended literature
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BLAŽEK, J. a kol. Algebra a teoretická aritmetika 1. Praha: SPN 1985..
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Haviár, M.: Algebra 1. OlBanská Bystrica, 2013. .
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