3 semester


The name of the discipline

Commutative algebra and elements of algebraic geometry


Curriculum Specialty

2nd year, Mathematics



Autumn (3rd) Semester


Semester credit hours



Full name of the lecturer

Beniash-Kryvets V.V.


Aims of the study of the discipline

To acquaint students with the basic concepts and methods of group theory and field theory, to study a number of important classes of groups and fields, as well as the most important classical results on groups and fields, to instill in students the ability to independently study educational and scientific literature in the field of mathematics, to expain some non-trivial constructions and techniques , used by working specialists.

As a result of studying the discipline the student should be able to:

  • use the Sylow theorems to describe the structure of finite groups;
  • decompose a finitely generated Abelian group into a direct product of primitive cyclic subgroups;
  • calculate the lower and upper Central series of the group;
  • build finite fields of a given order and perform calculations in them;
  • build simple extensions of a given field and perform calculations in them;
  • use the main theoretical results for solving computational problems related to groups, rings and fields.



algebra and number theory


The content of the discipline

  • Ideals and affine varieties.
  • Hilbert’s basis theorem. Groebner basises.
  • Exclusion theory.
  • Hilbert’s Nullstellensatz.
  • Applications of Groebner bases.
  • Polynomial and rational functions on a variety.
  • Morphisms of varieties.
  • The dimension of a variety.
  • Irreducible varieties. The decomposition into irreducible components.


The recommended literature

  1. Cox D., Little J., O’Shea D. Ideals, varieties and algorithms. New York: Springer-Verlag, 1997.
  2. Arzhancev I. Lections on Groebner basises. Moskau, 2002.
  3. Prasolov V.V. Polynomials. New York: Springer-Verlag, 2004.

Advanced reading:

  1. W.W. Adams, P. Loustaunau. An introduction to Gröbner bases. AMS, 1994.
  2. J. von zur Gathen, J. Gerhard. Modern computer algebra. Cambridge University Press, 1999.
  3. T. Becker, V. Weispfenning. Gröbner bases. A computational approach to commutative algebra. Springer-Verlag, 1993.


Methods of teaching

Verbal, visual, problem-based, practical, dialog-based and heuristic.


Language of education



Conditions (requirements), formative and summative assessment

  • check of individual tasks,
  • control work,
  • test.


System of assesment

control test