1 semester


The name of the discipline

Groups and fields theory


Curriculum Specialty

Master Program, first year,




Autumn 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

Homomorphisms of groups. Theorems on homomorphisms.

The most important parts of a group.

Actions of groups on sets.

Double cosets.

Sylow theorems.

Free abelian groups.

Finitely generated abelian groups.

Solvable groups.

Nilpotent groups.

Simple finite groups.

Free groups, subgroups of free groups.

Presentations by generators and relations.

Algebraic and transcendental field extensions.

Algebraically closed fields.

Separable extensions of fields.

Normal extensions of fields.

Finite fields.

Elements of Galois theory.


The recommended literature

  1. Vinberg E.B. A course in algebra. AMS, Graduate Studies in Mathematics, V: 56. 2003; 511 pp.
  2. Kostrikin A.I. Introduction to algebra. Springer-Verlag, 1982. 575 pp.
  3. Kargapolov M.I., Merzljakov J.I. Fundamentals of the theory of groups. Springer-Verlag, 1979. 203 pp.

Advanced reading:

  1. Lang S. Algebra. Springer-Verlag, 2002. 933 pp.
  2. Van der Waerden B.L.. Algebra, Vol. 1-2. Springer-Verlag, 2003.
  3. Kostrikin A.I. Exercises in algebra: A collection of exercises, in algebra, linear algebra and geometry. CRC Press, 1996. 474 pp.


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,

– tests.

examinations marks are given taking into account:

40% – semester work,

60% – oral answer in an examination


System of assesment

an examination