1 semester

1

Name of the course

Algebra and Number Theory (Part 1)

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Year of study, speciality

1, Mathematics and Information Technology (Web development and Internet technologies)

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Semester of study

1

4

Credits

4

5

Lecturer

Tikhonov S.V.

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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

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Prerequisites

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Contents

• Arithmetic of integers. Congruences
  Divisibility of integers and its properties. The theorem on division with remainder. The greatest common divisor. The Euclidean algorithm and the greatest common divisor as an integer linear combination of the two original numbers.
• Coprime integers, coprimality criterion. The least common multiple. Prime and composite numbers, the infinity of the set of prime numbers. The main theorem of arithmetic. Congruences and their properties.
• Residue classes. Operations on residue classes.
• Euler’s number-theoretic function, its multiplicativity. Theorems of Euler and Fermat.  Solution of linear congruences in one unknown. The Chinese remainder theorem.
• Algebraic operation, basic algebraic structures
• Properties of an algebraic operation: commutativity and associativity. Identity element and inverse elements with respect to an algebraic operation. Uniqueness of identity element and inverses.
• Definitions of groups, rings, fields. Examples. The ring of residue classes. Invertible residue classes. Finite fields .
• The field of complex numbers
• Definition of the field of complex numbers. Algebraic form of complex numbers. Complex conjugation. Complex plane. Module and argument of a complex number, their properties.
• Trigonometric form of a complex number. Operations on complex numbers in trigonometric form. De Moivre’s Formula and its application in real calculations.
• Geometric interpretation of operations with complex numbers. Extracting a root of a complex number. Roots of unity.
• Matrices and operations with them
• Matrix size. Types of matrices: square matrix, diagonal matrix, upper and lower triangular matrix, identity matrix, zero matrix, row and column vectors. Equality of matrices.
• Operations with matrices: addition and multiplication of matrices, matrix multiplication by a scalar, transpose. Properties of operations with matrices. Matrix polynomial.
• Permutations and substitutions
• Determination of permutations and substitutions, their number. Inversions, parity of permutation. Transpositions and cycles. Multiplication of permutations and its properties, symmetric group.
• Writing a permutation as a product of disjoint cycles and transpositions. Parity of a permutation.
• Determinants and their application
  Determinants of the second and third orders. The determinant of a square matrix of any order and its properties. The determinant of the transposed matrix.
• and algebraic complements. Laplace theorem. 
• Computing a determinant by row and column expansion.
• The determinant of a triangular matrix.  The determinant of a Vandermonde matrix. The determinant of the product of square matrices.
• The inverse of a matrix: the criterion of existence and methods of calculation. Complete linear group. Cramer’s theorem.
• Polynomials in one and several variables
  A ring of polynomials in one variable over a field. The degree of a polynomial and its properties. The theorem on division with remainder for polynomials. The greatest common divisor of polynomials, the Euclidean algorithm. Coprime polynomials. Irreducible polynomials. The theorem on decomposition of a polynomial into irreducible factors.
• The value of a polynomial at a point, a root of a polynomial. Bezout’s theorem and its corollaries. Horner’s scheme. The derivative of a polynomial and its properties. Multiplicity of a root of a polynomial. The fundamental theorem of algebra.
• The decomposition of a polynomial over the fields of complex and real numbers. Polynomials in n variables. Symmetric polynomials.

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Recommended Literature

1. Milovanov M.V, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 1. Minsk. .: Amalfea, 2001.
2. Milovanov M.V, Tolkachev M.M, Tyshkevich R.I, Fedenko A.S. Algebra and analytic geometry. V. 2. Minsk .: Amalfee, 2001.
3. Burdun A.A, Murashko E.A, Tolkachev M.M, Fedenko A.S. A collection of problems on algebra and analytic geometry. Moscow: University, 1999.
4. Monakhov V.S, Buzlanov A.V. Algebra and Number Theory: Workshop. Minsk: Izd.center of BSU, 2007.
5. Proskuryakov I.V. Collected problems on linear algebra. Moscow: Nauka, 1974.
6. Faddeev D.K, Sominsky I.S. Collection of problems on higher algebra. Moscow: Nauka, 1977.
7. Barkovich O.A. Algebra: tasks for practical studies and independent work.  Part 1. Introduction to algebra. Minsk: BSPU, 2005.
8. Barkovich O.A. Algebra: tasks 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.

Additional literature

1. Ireland K., Rosen M. Classical introduction to the modern theory of numbers. Moscow: Mir, 1987.
2. Van der Waerden B. L. Algebra. Moscow: Nauka, 1976.
3. Kostrikin A.I, Manin Yu.I. Linear Algebra and Geometry. Moscow: Nauka, 1983.
4. Kargapolov M.I, Merzlyakov Yu.I. Fundamentals of group theory. Moscow: Nauka, 1972.
5. Lang S. Algebra. Moscow: Mir, 1968.
6. Collection of problems in algebra. Ed. A.I. Kostrikin. Moscow: Nauka, 1987.

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Teaching methods

Verbal, visual, problematic, practical, dialog-heuristic.

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Language of teaching

Russian

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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.

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Form of current assessment

Examination, offset