# 1 semester

1

Name of the course

Algebra

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Year of study, speciality

1,

Mechanics and Mathematical Modeling

3

Semester of study

1

4

Credits

4

5

Lecturer

Ivanov K.A.

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Course objectives

To acquaint students with fundamental methods of general and linear 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:

perform operations with complex numbers in algebraic and trigonometric form, to extract roots of complex numbers, use de Moivre formula;

– calculate determinants;

– perform operations on matrices;

– solve systems of linear equations.

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Prerequisites

8

Contents

Arithmetic of integers, complex numbers

The theorem on division with remainder for integers. Euclidean algorithm. Definition of complex numbers, conjugate complex numbers.

Trigonometric form of a complex number, Muavr formula, geometry of operations on complex numbers.

N-th degree roots of a complex number, n-th degree roots of unity, primitive roots and their properties.

Matrices and operations on matrices

Rectangular matrices, equality of matrices, matrix addition and multiplication of a matrix by scalar, transposition of a matrix.

Matrix multiplication, associativity of matrix multiplication, the relationship between the operations of addition, multiplication and transpose of matrices.

Permutations, substitutions. Determinants and their application

The number of permutations of a finite set, the parity of the permutation, the number of even (odd) permutations of a finite set.

The number of substitutions of a finite set, the parity of the substitutions, the decomposition of the substitutions into the product of independent cycles. Definition of the determinant and its properties.

Laplace’s Theorem. Construction of the inverse matrix, Cramer rule.

Polynomials in one variable

The definition of a polynomial of one variable, equality of polynomials, the theorem of division with remainder,
Bezout theorem, the circuit of Horner.

Roots of polynomials, multiple roots, rational roots of polynomials with integer coefficients.

The main theorem of the algebra of complex numbers, the Vieta formulas, irreducible polynomials over Q, R, and C.

Algebraic operation, concept of a group, a ring, a field.

Definition of binary algebraic operation, neutral and symmetric elements. Definitions and examples of groups, rings, fields and their properties.

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### 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.

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Teaching methods

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

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Language of teaching

Russian

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Conditions (requirements), current control

– tests.

examinations marks are given taking into account:

40% – semester work,

60% – oral answer in an examination

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Form of current assessment

an examination, control test