Theory of Rings and Fields

Published

18 October 2026

Passport

  • Where and when. Semester IV of program 01.03.02 β€œApplied Mathematics and Informatics”, profile β€œMathematics”, in the curriculum for the 2025 intake.
  • Workload. 135 hours, 3 credits: 60 contact hours (30 lectures and 30 seminars), 45 hours of independent work and 30 hours for the exam.
  • Department. Department of Discrete Mathematics, according to the available program and the page of school-wide courses.
  • Lecturer. Not named in the open 2023 program that was found; this page should not be used as a timetable for a particular lecturer.
  • Final assessment. Exam in semester IV according to the curriculum for the 2025 intake.

Contents of the published program

Below is the complete list from the available program of the Department of Discrete Mathematics for spring 2023. This is not a claim that the 2025/26 program is unchanged, but it is the most detailed open primary document for the course.

  1. Rings, divisibility properties and Euclidean rings.
  2. Unique factorization in a Euclidean ring; primes in the Gaussian and Eisenstein integers.
  3. Pythagorean triples and the special case \(n=3\) of Fermat’s Last Theorem.
  4. Subrings, ideals, principal ideal rings, maximal and prime ideals, Noetherian rings and Hilbert’s basis theorem.
  5. The field of fractions, unique factorization in a polynomial ring over an integral domain, and Eisenstein’s criterion.
  6. Extensions of rings and fields, finite extensions and adjoining a root of an irreducible polynomial.
  7. Algebraic closure and the primitive element theorem.
  8. Ruler-and-compass constructions.
  9. Field homomorphisms and normal extensions.
  10. Galois extensions and the correspondence between subgroups and intermediate fields.
  11. Finite fields: uniqueness, subfields and the number of irreducible polynomials of a given degree.

Literature in the 2023 program

  • E. B. Vinberg, A Course in Algebra, 2nd ed., 2001.
  • A. I. Kostrikin, Introduction to Algebra, in three parts, 2001.
  • M. Artin, Algebra, 2011.
  • R. B. J. T. Allenby, Rings, Fields and Groups, 1991.

Assessment

No open grading scheme of the Department of Discrete Mathematics has been found specifically for this discipline. Numerical grading rules are not given on this page.

Sources and currency

Program β€œTheory of Rings and Fields”, 2023, combined program, curriculum 01.03.02, 2025 intake.