Mathematik

I. The content of science

The goal of teaching science is to introduce and teach students algebraic concepts on a scientific basis and to lay the foundation for a correct analysis of changes in the development of nature and society by revealing their many applications, to provide students with information about mathematical theories that are already familiar to them, especially linear algebra, basic algebraic structures, number theory, and polynomial theory, to develop their logical and mathematical thinking based on the theoretical knowledge they have acquired, and to form written and oral logical speech.

The task of the subject is to systematically expand the basic concepts of mathematics in general secondary, secondary specialized, and vocational education along with scientific and theoretical deepening, to reveal the role of algebra and number theory in the formation of students' worldviews in the study of society and existence, to teach students the theoretical foundations of the algebra and number theory course, to form in them the necessary skills and qualifications for mastering the algebra and number theory course, to introduce students to the algebra and number theory course, and to teach them to work independently with textbooks and other scientific literature.

II. Main theoretical part (lectures)

The science includes the following topics:

Topic 1 : Sets and operations on them.

Set , set element. Equality of sets . Partial set . Empty set. Universal set . Operations on sets and their main properties. Complement of a set .

Topic 2: Binary relationships . Equivalence relation.

Binary relations. Definition of binary relations and value domains. Inversion of a binary relation.

Topic 3: Reflections.

Reflection. Types of reflections (injective, surjective, bijective reflections). Composition of reflections. Inverse reflection.

Topic 4. Small-order determinants.

Second and third order determinants and methods for calculating them .

Topic 5: Substitutions and substitutions.

Properties of permutations and substitutions. Odd and even permutations. Multiplication of substitutions.

Topic 6. Matrices and operations on them .

Types of matrices. Addition of matrices , multiplication of a matrix by a number, operations of matrix multiplication and their properties. Transpose of a matrix.

Topic 7. Determinants of order n and their properties.

n-order determinants and their properties.

Topic 8. Minors and algebraic complements .

The p-order minor of a determinant. The algebraic complement of an arbitrary element of a determinant .

Topic 9: Laplace's theorem.

Expanding the determinant of a matrix by row or column elements. Laplace's theorem.

Topic 10 : Inverse matrix. Additional properties of the determinant.

Inverse matrix. Inverse matrix s harts. Calculating the inverse matrix. Additional properties of the determinant.

Topic 11 : Systems of linear equations and the Gauss method for solving them.

Systems of linear equations. Equivalence of systems of linear equations. Gauss's method.

Topic 12 : Cramer's method for solving systems of linear equations.

Cramer's method for solving a system of linear equations.

Topic 13. Dimensionless vector space. Linearly dependent and linearly free vectors .

linearly dependent and linearly independent vectors .

Topic 14. Theorems about linearly dependent and linearly free vectors.

Theorems about linearly dependent and linearly free vectors . Basis and color of a vector system.

Topic

Matrix color. Nonhomogeneous linear equations system . Kronecker-Capelli theorem.

Topic 16 . System of homogeneous equations. Fundamental solution.

Homogeneous systems and methods for solving them. Fundamental systems of solutions.

Topic 17. Complex numbers and operations on them. Geometric representation and trigonometric form of complex numbers.

Operations on complex numbers and their properties. Co-occurring complex numbers and their properties. The modulus of a complex number and its properties.

Topic 18: Moivre's formula, extracting roots from complex numbers. Roots of one.

Moivre's formulas. Roots of one to the power of p. Roots of an arbitrary complex number to the power of p.

Topic 19. Polynomials with one unknown. Gorner scheme. Bezu's theorem.

Polynomials and operations on them. Gorner scheme. Bezu's theorem.

20. Remainder. The Eq. of polynomials.

Dividing polynomials with remainders. Greatest common divisor. Euclidean algorithm. Unreducible polynomials.

21. Rational fractions and their expansion into simplest fractions.

Rational fractions and their expansion into simplest fractions.

Topic 22. Solving third and fourth degree algebraic equations.

Cardano's formula and Ferrari's method.

23. Root limits. Sturm's theorem.

Root limits. Sturm theorems.

24. Linear spaces.

Linear spaces. Dimension and basis of linear spaces.

25. Linear partial space. Sum and intersection of partial spaces .

Linear partial space. Direct sum of partial spaces . Sum and intersection of partial spaces .

26. Euclidean space.

Euclidean space. Cauchy-Buniakovsky inequality. Orthogonal and orthonormal systems.

Topic 27. Isomorphism of linear and Euclidean spaces.

Orthogonal expansion process . Isomorphism of linear and Euclidean spaces.

28. Orthogonal complement.

Orthogonal complement and orthogonal projection.

29. Linear, bilinear, and quadratic forms .

Linear, bilinear, and quadratic forms. Transformation of a matrix of a bilinear form when the basis changes.

Topic 30. Quadratic form .

Converting a quadratic form to canonical form methods.

31. Law of inertia.

Positive definite quadratic forms. Law of inertia.

Topic 32. Scalar multiplication in complex vector spaces .

Complex Euclidean spaces. Linear forms in complex space.

Topic

Linear permutations and their matrices. Operations on linear permutations.

Topic 34. Reverse substitution .

Reverse replacement. Image and kernel of linear permutations . Connection between linear permutation matrices in different bases.

Topic 35. Invariant part spaces .

Invariant subspaces. The x os number and the eigenvectors i of a linear permutation .

36. Q joint replacement.

substitution for a given substitution .

Topic 37. Self-joint and unitary substitutions.

Self-combined permutations and their canonical form. Unitary permutations, their eigenvalues and canonical representation.

38. Commutative and normal permutations.

Commutative permutations. Normal permutations and their canonical representation.

Topic 39. Jordan normal form of a linear permutation matrix .

Jordan normal form of a linear permutation matrix .

Topic 40. Converting a linear permutation matrix to Jordan normal form .

Converting a linear permutation matrix to Jordan normal form .

Topic 41 : Division symbols. Common divisor and multiple of numbers.

Common divisor and multiple of numbers. Remainder .

Topic 42: Continued and proper fractions, properties.

Representation of rational numbers as finite continued fractions. Proper fractions and their basic properties.

Topic 43: Comparisons and their properties.

Comparison and its properties. Classes of discounts by module. Complete system of discounts and its properties. The given system of discounts and its properties.

Topic 44 : Multiplicative functions. Euler and Fermat's theorems. Multiplicative functions. Euler function. Multiplicativeness of the Euler function. Formula for calculating the Euler function. Euler's theorem. Fermat's theorem.

Topic 45: First-order comparisons. Chinese remainder theorem.

Theorem on the number of solutions of first-order equations with one unknown. Methods for solving first-order equations with one unknown. Chinese theorem on residues.

Topic 46. Semigroups. Monoids. Groups.

Binary operations. Semigroups. Monoids. Groups. Abelian group.

Topic 47. Symmetric and sign-commuting groups.

Symmetric and sign-commuting groups

48 - topic . Part groups. Cyclic groups.

Part Groups. Forming element. Cyclic groups.

49. Right and left neighbor classes. Lagrange's theorem.

Right and left adjoint classes and their properties. Concepts of finite group and index. Lagrange's theorem.

Topic 50. Normal partial groups and factor groups.

Normal division group and its properties. Relations between adjacent classes. Factor groups.

Topic 51. Homomorphisms and isomorphisms of groups.

Homomorphisms of a group. Kernel of homomorphism. Basic concepts of epimorphism, monomorphism and isomorphism.

52. Properties of homomorphisms and isomorphisms. Cayley's theorem.

Properties of homomorphisms and isomorphisms. Cayley's theorem.

Topic 53. Dihedral and quaternion groups.

dihedral and quaternion groups.

54. Theorems about group homomorphisms.

The fundamental theorem of homomorphism. The first, second, and third theorems on isomorphisms.

Topic 55. Group automorphisms and internal automorphisms

Automorphisms of a group and the group defined between them. Inner automorphisms and their properties.

Topic 56. The influence of the group on the set.

of the group on the set . Orbit. Stationary part of groups. The overlap of the orbital length with the stationary group index.

Topic 57. Rings and their types .

Rings and their types, the domain of integrity . Nilpotent and idempotent elements of a ring .

Topic 58. Characteristics of rings. Boolean and regular rings.

Ring characteristic. Properties of Boolean and regular rings .

Topic 59. Partial rings.

Partial rings and theorems about them.

60. Ideals of a ring. Factor rings. Ring of prime ideals.

Right, left and dual ideals of a ring. Factor rings.

Topic 61. Homomorphisms and isomorphisms of rings

Ring homomorphisms and their properties. Theorems about ring homomorphisms .

Topic 62. Nilpotent and primary ideals .

Nilpotent and prime ideals . Prime and irreducible elements of a ring.

Topic 63. Maximal and primary ideals.

Maximal and primary ideals. Ring radical.

Topic 64. Bodies and areas.

Definitions of body and area and their examples.

Topic 65. Field characteristics.

Field characteristics. Fields with finite characteristics.

Topic 66. Field extensions.

Extensions of a field. Separable and normal extensions of a field.

Topic 67. Finite fields.

Finite fields and their examples.

III. Instructions and recommendations for practical exercises.

The following topics are recommended for practical training:

Topic 1. Sets and operations on them.

Topic 2. Binary relationships. Equivalence relation.

Topic 3. Reflections.

Topic 4. Small-order determinants. Substitutions and substitutions.

Topic 5. Matrices and operations on them .

Topic 6. Determinants of order n and their properties. Minors and algebraic complements .

Topic 7. Laplace's theorem. Inverse matrix. Additional properties of the determinant.

Topic 8. Systems of linear equations and the Gauss method for solving them. Cramer's method for solving systems of linear equations.

Topic 9: Dimensionless vector space. Linearly dependent and linearly independent vectors . Theorems about linearly dependent and linearly independent vectors.

Topic 10: Matrix color. System of non-homogeneous linear equations . Kronecker-Capelli theorem. System of homogeneous equations. Fundamental solution.

Topic 11: Complex numbers and operations on them. Geometric representation and trigonometric form of complex numbers.

Topic 12: Moivre's formula, extracting roots from complex numbers. Roots of one.

13. Polynomials with one unknown. Gorner scheme. Bezu's theorem. Remainder. Eq. of polynomials.

14. Rational fractions and their expansion into simplest fractions.

Topic

Topic 16. Linear spaces. Linear partial space. Sum and intersection of partial spaces .

Topic 17. Euclidean space. Isomorphism of linear and Euclidean spaces.

Topic 18. Orthogonal complement.

19. Linear, bilinear and quadratic forms . Quadratic form .

Topic 20. Law of inertia.

Topic 21. Scalar product in complex vector spaces . Linear permutations .

Topic 22. Reverse substitution .

Topic 23. Invariant part spaces .

Topic 24. Q- joint permutation. Self-joint and unitary permutations.

Topic 25. Commutative and normal permutations.

Topic 26. Jordan normal form of a linear permutation matrix .

Topic 27. Converting a linear permutation matrix to Jordan normal form .

28 : Division signs. Common divisor and multiple of numbers. Continued and proper fractions, properties.

29 : Comparisons and their properties.

30 : Multiplicative functions. Euler's and Fermat's theorems. First-order comparisons. Chinese remainder theorem.

Topic 31. Semigroups. Monoids. Groups.

Topic 32. Symmetric and sign-commuting groups.

Topic 33 Part​ groups. Cyclic groups.

Topic 34. Right and left adjoint classes. Lagrange's theorem. Normal partial groups and factor groups.

Topic 35. Homomorphisms and isomorphisms of groups.

Topic 36. Properties of homomorphisms and isomorphisms. Cayley's theorem.

Topic 37. Dihedral and quaternion groups. Theorems on group homomorphisms.

Topic 36. Group automorphisms and inner automorphisms

Topic 39. The effect of a group on a set.

Topic 40. Rings and their types . Characteristics of rings. Boolean and regular rings.

Topic 41. Partial rings. Ideals of a ring. Factor rings. Ring of prime ideals.

Topic 45. Homomorphisms and isomorphisms of a ring

Topic 43. Nilpotent and primary ideals . Maximal and primary ideals.

Topic 44. Bodies and areas. Characteristics of areas.

45. Field extensions. Finite fields.

IV. Independent learning and independent work

When preparing an independent work , taking into account the characteristics of the subject " Algebra and Number Theory " , the student is recommended to use the following forms:

• Preparation for practical training ;
• Studying science chapters and topics from textbooks and study guides ;
• Mastering the lecture part of the subject using handouts ;
• Work on scientific sections or topics using specialized literature ;
• In-depth study of sections and topics of science related to the student's educational, scientific and research work ;
• Use of distance learning and so on.

is carried out by students writing abstracts , drawing and analyzing diagrams , filling out tables , and collecting additional information on topics given in advance .

Recommended topics for independent study:

Recommended topics for independent work:

1. Axiomatic construction of determinant theory.
2. Fundamental theorem of algebra.
3. Third and fourth order equations.
4. Sturm's theorem and its applications.
5. Polynomials with rational coefficients.
6. Results that follow from the fundamental theorem
7. Some applications of complex number theory
8. Dividing a polynomial into multiple factors
9. Polynomial root boundaries
10. Fundamental solution.
11. Algebraic number field.
12. Orthogonal complement and its properties. Methods for finding a basis in an orthogonal complement space .
13. Calculating orthogonal constituents and orthogonal projections.
14. Joint and self-joint linear permutations. Their matrices.
15. Finding characteristic roots of symmetric linear permutations .
16. Right and left adjoint classes, Lagrange's theorem.
17. Normal divisors. Factor groups.
18. Homomorphisms and isomorphisms of a group.
19. Properties of homomorphisms and isomorphisms. Cayley's theorem.
20. The influence of the group on the collection.
21. Orbit. Stationary part groups.
22. The overlap of the orbital length with the stationary group index.
23. The concept of a ring. Types of rings. Partial rings. Factor rings.
24. Ideal types. Nyoter and Artin rings.
25. Field extension.
26. Galois field. Finite and algebraic extensions.
27. The level of expansion.
28.  group founders.
29.  Q is a local cyclic group of .
30.  Homomorphisms of algebras .

Note: Based on the volume of independent study hours, independent study topics are formed from these topics in the work program.

    It is recommended that students prepare and present abstracts on topics that are being studied independently.

V. Results of science teaching (developed competencies)

mastering the subject , the student will:

• To have an idea and knowledge of the basic concepts and foundations of algebra and number theory, the traditions of the development of the science, and its place among mathematical sciences;
• specific features of the basic concepts and foundations of algebra and number theory , the fundamental issues of the science, and examples and problems regarding its place among mathematical sciences, and to have the skills to use them;
• The student must be able to apply the principles of algebra and number theory to analyze problems and solve problems that arise when solving these problems.

VI. Educational technologies and methods:

• Lectures;
• Interactive case studies;
• Presentations;
• Working in groups;
• Teamwork and advocacy;

VII. Requirements for obtaining a loan:

Fully master the theoretical and methodological concepts of the subject, be able to correctly reflect the results of the analysis, conduct independent observations of the processes being studied, and complete the tasks assigned in the current and intermediate control forms, and submit the final control work.

Main literature

1. D. S. Malik , John N. Mordeson, MKSen, Fundamentals of Abstract Algebra, 1997, P. 636.

2. J.​ Khojiev , A.​ S.​ Feinleib , Algebra and numbers theory course . T .: " Uzbekistan " , 2001. 304 p .

3. Sh. A. Ayupov , BAOMirov, Algebra and number theory; T.: "Tafakkur-bo'stoni" publishing house, 2019.

4. R.​ N.​ N. Azarov , B.​ T.​ T oshp o ' Latov , A. D.​ D usumbetov . Algebra​ and numbers theory T., Teacher . 1 - part , 1993. 320 p .

5. Yunusov A. , Yunusova D. Numerical systems . T. , Finance and Economics , 2008. 116 p. ( textbook ) .

Additional literature

1. R.N. Nazarov, B.T. Toshpolatov, A.D. Dusumbetov. Algebra and number theory T., Teacher. Part 2, 1995.

2. Yunusova D., Yunusov A. Algebra and number theory. A collection of examples and exercises based on module technology . Textbook . T., "Ilm Ziyo", 2009.

A set of control tasks in algebra and number theory based on module technology . TDPU, 2004.

Internet resources

1. www.tdpu.uz

2. www.pedagog.uz

3. www.edu.uz

4. www.nadlib.uz ( A. Navoi University UzMK )