3 semester


Name of the course

Algebra and Number Theory (Part 3)


Year of study, speciality

2, Mathematics (Science-Industrial Activity)


Semester of study







Benyash-Krivets V.V.


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.

The score on the exam is set taking into account:

  • 40% – work in the semester,
  • 60% – oral answer in the exam.


Form of current assessment

an examination