MU25015 Chapters in Algebra

Mathematical Institute in Opava
Summer 2022

The course is not taught in Summer 2022

Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
doc. RNDr. Michal Marvan, CSc. (lecturer)
RNDr. Hynek Baran, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Michal Marvan, CSc.
Mathematical Institute in Opava
Prerequisites (in Czech)
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
Chapters in algebra cover advanced algebraic topics, appropriate for all graduates of mathematics.
  • 1. Integral domains, divisibility, Euclidean domains, principal ideal domains, unique factorization domains, field of fractions over an integral domain, resultant and discriminant.
    2. Algebraic and transcendental extensions, algebraic number fields.
    3. Systems of polynomial equations and their solutions, polynomial ideals, Hilbert's basis theorem.
    4. Groebner bases, Buchberger algorithm, arithmetics in quotient rings of polynomial rings.
    5. Algebraic varieties, radical ideals, sums, products and intersections of algebraic varieties, irreducible varieties and prime ideals.
    6. Groups, algebras and their representations, group modules, irreducible representations.
    required literature
  • J. Blažek, M. Koman, B. Vojtášková. Algebra a teoretická aritmetika. II. díl. Praha, 1985. info
  • G.D. Mostow, J.H. Sampson, J.-P. Meyer. Fundamental structures of algebra. New York, 1963. info
    recommended literature
  • D.A. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms. 2015. ISBN 978-3-319-16721-3. info
  • S. Lang. Algebraic structures. Reading, 1967. info
Language of instruction
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
Teacher's information
The course comprises lectures and tutorials. To pass the course, the first step is to earn credit for tutorials (by earning 70% on a written test). The final exam, which consists of a written and on oral part, tests theoretical knowledge and understanding of the subject, including proofs.

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