3 semester


Name of the course

Algebra and Number Theory (Part 3)


Year of study, speciality

2, Mathematics (Economic Activity)


Semester of study







Voronovich I.I.


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:

  • reduce a bilinear and quadratic form to the canonical form, to the normal form on R and C;
  • apply Sylvester’s criterion;
  • set the scalar product in Euclidean space;
  • calculate the lengths of vectors and angles between them;
  • find the orthogonal projection and orthogonal component of the vector;
  • apply the process of Gram-Schmidt orthogonalization;
  • find an orthonormal basis, an orthogonal complement to the subspace;
  • find an adjoint operator;
  • reduce the orthogonal and self-adjoint operator to the canonical form;
  • determine whether a given subset is a subgroup in a group, subring or an ideal in a ring, subfield in a field;
  • carry out computations in the quotient group, quotient ring.



Algebra and Number Theory (Part 1-2)



Bilinear and quadratic forms.

  • Bilinear form on a vector space, its matrix. The change of the matrix of a bilinear form with the change of basis, rank of a form. Symmetric and skew-symmetric bilinear forms, their matrices.
  • Operations on bilinear forms, the space of bilinear forms and its isomorphism to the space of square matrices. Quadratic form and its matrix, existence and uniqueness of polar bilinear form.
  • Canonical form of a bilinear and quadratic form. Quadratic Forms Lagrange method to reduce a quadratic form to the canonical form. A normal form of a real and complex quadratic form.
  • The law of inertia of real quadratic forms. Positive- and negative-definite quadratic forms, Sylvester’s criterion.

Euclidean spaces.

  • Definition of an Euclidean space. The length of the vector, the angle between the vectors. The Cauchy-Bunyakovsky-Schwarz Inequality.
  • Orthonormal families of vectors, orthonormal bases. Gram–Schmidt process. Orthogonal complement to the subspace.
  • Decomposition of a space into a direct sum of a subspace and its orthogonal complement.

Linear operators of Euclidean spaces.

  • An adjoint operator, its existence and properties. Invariant subspaces for the adjoint operators.
  • Orthogonal operators, canonical forms of their matrices. A selfadjoint operator.
  • The existence of an orthogonal transformation reducing a real quadratic form to a diagonal form.

Introduction to the theory of groups.

  • The definition of groups, subgroups, examples. Homomorphism, isomorphism, automorphism. The order of the group element.
  • A cyclic subgroup. Cyclic groups, their classification.
  • Cosets of a subgroup, the index of the subgroup. Lagrange’s theorem and corollaries.
  • A normal subgroup. A quotient group. The main theorem on homomorphisms of groups. A direct product of groups.

Introduction to the theory of rings and fields.

  • The definition of a ring, of a subring, of a field, of a subfield, examples. A homomorphism, an isomorphism of rings, the kernel of a homomorphism.
  • Ideals of rings. A quotient ring. The main theorem on homomorphisms for rings. A direct product of rings.
  • The characteristic of the field. Simple fields.
  • The degree of a field extension, finite extensions. The multiplicativity formula for degrees.
  • Algebraic and transcendental elements. Simple field extensions. Algebraically closed fields, algebraic closure.


Recommended Literature

  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:

  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.


Teaching methods

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


Language of teaching



Conditions (requirements), current control

  • check of individual tasks,
  • tests.

examinations marks are given taking into account:

  • 40% – semester work,
  • 60% – oral answer in an examination


Form of current assessment

an examination