4 semester

Name of the course

Additional chapters of algebra

Year of study, speciality

Mathematics (Science-Industrial Activity)

Semester of study

Credits

Lecturer

Benyash-Krivets V.V.

Course objectives

To acquaint students with fundamental methods of general algebra, with basic algebraic structures – groups, rings and fields; to create a base for studying basic concepts and methods of modern mathematics; to form mathematical thinking among students; to acquaint with methods of mathematical proofs; to study algorithms for solving specific mathematical problems. To instill in students the ability to independently study educational and scientific literature in the field of mathematics.

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

– determine whether a given subset is a subgroup in a group, a subring or an ideal in a ring, a subfield in a field;

– to perform calculations in a quotient group, quotient ring;

– build finite fields of a given order and perform calculations in them;

– apply the most important theoretical results to solving computational problems related to groups, rings and fields.

Prerequisites

Contents

Introduction to group theory.

The definition of groups, subgroups, examples.

A homomorphism, an isomorphism, an automorphism. The order of the element.

Generating sets. Cyclic groups, their classification.

Cosets for a subgroup, the index of the subgroup. Lagrange theorem, Lagrange decomposition, a consequence on the order of an element.

Normal subgroups. A quotient group.

The kernel and the image of a homomorphism. The first (main) theorem on homomorphisms, its application to the calculation of the quotient group.

The relationship between the subgroups of a quotient group and intermediate subgroups. The second and third theorems on homomorphisms.

Direct product of groups and decomposition of a group into a direct product of its subgroups. Free abelian groups.

A theorem on the structure of a finitely generated abelian group. A center and a derived subgroup. The criterion of commutativity of a quotient group.

Introduction to ring theory.

The definition of a ring, of a subring, of a field, of a subfield, examples. The multiplicative group of a ring.

The homomorphism, isomorphism of rings, the kernel of a homomorphism. Ideals of rings. A quotient ring.

The main theorem on homomorphisms for rings. Principal ideal. Principal ideal ring. Ideals in and .

Maximal ideals and their corresponding quotient rings. The inner and outer direct products of rings. The structure of the ring and arithmetic consequences.

Introduction to field theory.

The characteristic of the field. Simple fields. The degree of an extension, finite extensions. The multiplicativity of a degree.

Algebraic and transcendental elements. The minimal polynomial of an algebraic element. Algebraic extensions, algebraicity of finite extension.

Simple field extensions. Algebraically closed fields, the algebraic closure. The field of fractions of a ring without zero divisors.

The number of elements of a finite field. The theorem on the existence and uniqueness of a field containing elements. Subfields of a finite field. The multiplicative group of a finite field. Irreducible polynomials over a finite field.

RecommendedLiterature

1. Milovanov M.V, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 1. Minsk: Amalfeya, 2001.

2. Milovanov M.V, Tolkachev M.M, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 2. Minsk: Amalfeya, 2001.

3. Burdun A.A, Murashko E.A, Tolkachev M.M, Fedenko A.S. A collection of problems on algebra and analytic geometry. Minsk: Universitetskoe, 1999.

4. Monakhov V.S, Buzlanov A.V. Algebra and Number Theory: Workshop. Minsk: Izd. center of BSU, 2007.

5. Proskuryakov I.V. Problems in linear algebra. Moscow: Nauka, 1978.

6. Faddeev D.K, Sominsky I.S. Collection of problems on higher algebra. Moscow: Nauka, 1977.

7. Barkovich O.A. Algebra: problems for practical studies and independent work. Part 1. Introduction to algebra. Minsk: BSPU, 2005.

8. Barkovich O.A. Algebra: problems for practical studies and independent work. Part 2. Linear algebra. Minsk: BSPU, 2006.

9. Kostrikin A.I. Introduction to Algebra. V. 1-3. Moscow: Phys.-Math. literature, 2000-2001.

10. Gelfand I.M. Lectures on linear algebra. Moscow: MTsNMO, 1998.

11. Kurosh A.G. The course of higher algebra. Moscow: Nauka, 1965 (and later editions).

12. Maltsev I.M. Fundamentals of linear algebra. Moscow: Nauka, 1970.

13. Faddeev D.K. Lectures on algebra. Moscow: Nauka, 1984.

14. Vinberg E.B. Course of algebra. Moscow: Factorial-press, 2001.

15. Vinogradov I.M. Fundamentals of number theory. Moscow: Nauka, 1976.

Advanced reading:

16. Ireland K., Rosen M. Classical introduction to the modern theory of numbers. Moscow: Mir, 1987.

17. Van der Waerden B. L. Algebra. Moscow: Nauka, 1976.

18. Kostrikin A.I, Manin Yu.I. Linear Algebra and Geometry. Moscow: Nauka, 1983.

19. Kargapolov M.I, Merzlyakov Yu.I. Fundamentals of group theory. Moscow: Nauka, 1972.

20. Lang S. Algebra. Moscow: Mir, 1968.

21. Collection of problems in algebra. Ed. A.I. Kostrikin. Moscow: Nauka, 1987.

Teaching methods

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

Language of teaching

Russian

Conditions (requirements), current control

– check of individual tasks,

– control work,

– test.

Form of current assessment

control test