1

Name of the course

Algebra and Number Theory (Part 2)

2

Year of study, speciality

1, Mathematics (Research Activities and Production)

3

Semester of study

2

4

Credits

4

5

Lecturer

Tikhonov S.V.

6

Course objectives

Teaching students fundamental methods of general algebra, linear algebra, number theory; acquaintance with basic algebraic structures – groups, rings and fields; creation of a base for studying basic concepts and methods of modern mathematics; formation of mathematical thinking among students; familiarity with methods of mathematical evidence; studying algorithms for solving specific mathematical problems. Teaching 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:
 perform operations with complex numbers in algebraic and trigonometric form, extract roots of complex numbers, apply De Moivre’s formula;
 calculate determinants;
 perform operations with matrices;
 solve systems of linear equations;
 find a basis of a vector space, the sum and intersection of subspaces, the coordinates of a vector in a given basis, find the rank of a matrix and system of vectors;
 find the eigenvalues and eigenvectors of a matrix and linear operator;
 reduce a quadratic form to the canonical form;
 reduce the matrix of an orthogonal operator to the canonical form;
 find an orthonormal basis, an orthogonal complement of a subspace;
determine whether a given subset is a subgroup of a group, a subring or an ideal of a ring, a subfield of a field;
 perform calculations in the factor group, factor ring

7

Prerequisites

Algebra and Number Theory (Part 1)

8

Contents

 Vector spaces
 Definition and examples. System of generators, finitedimensional spaces.
 Linear independence of vectors. Steinitz exchange lemma. Basis, dimension.
 Coordinates of a vector, their change when the basis is changed. Transition matrix, transformation of the coordinates of a vector. Subspace, its dimension.
 The rank of a system of vectors. The rank of a matrix.
 The sum and intersection of subspaces, their dimensions. The direct sum of subspaces.
 Systems of linear equations.
 Matrix form of a linear system. The Gauss method. KroneckerCapelli theorem. Homogeneous systems, the condition for existence of a nontrivial solution.
 A fundamental system of solutions. Connection between solutions of homogeneous and nonhomogeneous systems.
 The subspace defined by a system of linear equations.
 Differentiable functions in several variables
 Linear map, its core and image. Rank and defect. Algebraic operations on linear maps: sum, multiplication by a constant, composition.
 Linear operator and its matrix, its change when the basis is changed.
 Matrix of composition and sum of linear operators.
 The space of linear operators and its connection with the space of matrices. Conditions for the invertibility of an operator.
 Invariant subspaces. Eigenvectors and eigenvalues. Normal forms of matrices.
 Restriction of an operator to an invariant subspace.
 The matrix of an operator with an invariant subspace, under decomposition of the space into a direct sum of invariant subspaces.
 Characteristic polynomial of an operator and matrix.
 The HamiltonCayley theorem. An operator having a diagonal matrix in some basis; a condition of diagonalizability. Jordan matrix.
 A theorem on the existence of a Jordan normal form of a matrix. Algorithm of reduction to the Jordan normal form. Frobenius normal form.

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Literature 
 Milovanov M.V, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 1. Minsk. .: Amalfea, 2001.
 Milovanov M.V, Tolkachev M.M, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 2. Minsk .: Amalfee, 2001.
 Burdun A.A, Murashko E.A, Tolkachev M.M, Fedenko A.S. A collection of problems on algebra and analytic geometry. Moscow: University, 1999.
 Monakhov V.S, Buzlanov A.V. Algebra and Number Theory: Workshop. Minsk: Izd.center of BSU, 2007.
 Proskuryakov I.V. Collected problems on linear algebra. Moscow: Nauka, 1974.
 Faddeev D.K, Sominsky I.S. Collection of problems on higher algebra. Moscow: Nauka, 1977.
 Barkovich O.A. Algebra: tasks for practical studies and independent work. Part 1. Introduction to algebra. Minsk: BSPU, 2005.
 Barkovich O.A. Algebra: tasks for practical studies and independent work. Part 2. Linear algebra. Minsk: BSPU, 2006.
 Kostrikin A.I. Introduction to Algebra. V. 13. Moscow: Phys.Math. literature, 20002001.
 Gelfand I.M. Lectures on linear algebra. Moscow: MTsNMO, 1998.
Kurosh A.G. The course of higher algebra. Moscow: Nauka, 1965 (and later editions).
 Maltsev I.M. Fundamentals of linear algebra. Moscow: Nauka, 1970.
 Faddeev D.K. Lectures on algebra. Moscow: Nauka, 1984.
 Vinberg E.B. Course of algebra. Moscow: Factorialpress, 2001.
 Vinogradov I.M. Fundamentals of number theory. Moscow: Nauka, 1976.
Additional literature
 Ireland K., Rosen M. Classical introduction to the modern theory of numbers. Moscow: Mir, 1987.
 Van der Waerden B. L. Algebra. Moscow: Nauka, 1976.
 Kostrikin A.I, Manin Yu.I. Linear Algebra and Geometry. Moscow: Nauka, 1983.
 Kargapolov M.I, Merzlyakov Yu.I. Fundamentals of group theory. Moscow: Nauka, 1972.
 Lang S. Algebra. Moscow: Mir, 1968.
 Collection of problems in algebra. Ed. A.I. Kostrikina. Moscow: Nauka, 1987.

10

Teaching methods

Verbal, visual, problematic, practical, dialogheuristic.

11

Language of teaching

Russian

12

Conditions (requirements), current control

 verification of individual tasks,
 colloquium,
 test.
The score on the exam is set taking into account:
 40% – work in the semester,
 60% – oral answer in the exam.

13

Form of current assessment

Examination, offset
