## Higher Algebra – Kurosh

In this post we see Higher Algebra by A. Kurosh.

The education of the mathematics major begins with the
study of three basic disciplines: mathematical analysis, analytic
geometry and higher algebra. These disciplines have a number of
points of contact, some of which overlap; together they constitute
the foundation upon which rests the whole edifice of modern
mathematical science.

Higher algebra – the subject of this text – is a far-reaching and
natural generalization of the basic school course of elementary
algebra. Central to elementary algebra is without doubt the problem
of solving equations. The study of equations begins with the very
simple case of one equation of the first degree in one unknown. From
there on, the development proceeds in two directions: to systems of
two and three equations of the first degree in two and, respectively,
three unknowns, and to a single quadratic equation in one unknown and
also to a few special types of higher-degree equations which readily
reduce to quadratic equations (quartic equations, for example). Both
trends are further developed in the course of higher algebra, thus
determining its two large areas of study. One – the foundations of
linear algebra – starts with the study of arbitrary systems of
equations of the first degree (linear equations). When the number of
equations equals the number of unknowns, solutions of such systems
are obtained by means of the theory of determinants.

The second half of the course of higher algebra, called the algebra
of polynomials, is devoted to the study of a single equation in one
unknown but of arbitrary degree. Since there is a formula for solving
quadratic equations, it was natural to seek similar formulas for
higher-degree equations. That is precisely how this division of
algebra developed historically. Formulas for solving equations of
third and fourth degree were found in the sixteenth century. The
search was then on for formulas capable of expressing the roots of
equations of fifth and higher degree in terms -of the coefficients of
was futile, though it continued up to the beginning of the nineteenth
century, when it was proved that no such formulas exist and that for
all degrees beyond the fourth there even exist specific examples of
equations with integral coefficients whose roots cannot be written

This book was translated from the Russian by George Yankovsky. The  book was published by first Mir Publishers in 1972, with reprints in  1975, 1980 and 1984. The book below is from the 1984 reprint.

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Introduction 7
Chapter 1.
Systems of linear equations. Determinants 15

1. The Method of Successive Elimination of Unknowns 15
2. Determinants of Second and Third Order. 22
3. Arrangements and Permutations 27
4. Determinants of nth Order 36
5. Minors and Their Cofactors 43
6. Evaluating Determinants 46
7. Cramer’s Rule 53

Chapter 2.
Systems of linear equations ( general theory) 59

8. n-Dimensional Vector Space 59
9. Linear Dependence of Vectors 62
10. Rank of a Matrix 69
11. Systems of Linear Equations. 76
12. Systems of Homogeneous Linear Equations 82

Chapter 3.
The algebra of matrices 87

13. Matrix Multiplication 87
14. Inverse Matrices 93
15. Matrix Addition and Multiplication of a Matrix by a Scalar 99
16. An Axiomatic Construction of the Theory of Determinants 103

Chapter 4.
Complex numbers 110

17. The System of Complex Numbers 110
18. A Deeper Look at Complex Numbers 112
19. Taking Roots of Complex Numbers 120

Chapter 5.
Polynomials and their roots 126

20. Operations on Polynomials 126
21. Divisors. Greatest Common Divisor 131
22. Roots of Polynomials. 139
23. Fundamental Theorem 142
24. Corollaries to the Fundamental Theorem 151
25. Rational Fractions 156

Chapter 6.

26. Reducing a Quadratic Form to Canonical Form 161
27. Law of Inertia. 169
28. Positive Definite Forms 174

Chapter 7
Linear spaces 178

29. Definition of a Linear Space. An Isomorphism 178
30. Finite-Dimensional Spaces. Bases 182
31. Linear Transformations 188
32. Linear Subspaces. 195
33. Characteristic Roots and Eigenvalues 199

Chapter 8
Euclidean spaces204
34. Definition of a Euclidean Space. Orthonormal Bases 204
35. Orthogonal Matrices, Orthogonal Transformations. 210
36. Symmetric Transformations. 215
37. Reducing a Quadratic Form to Principal Axes. Pairs of Forms 219

Chapter 9.
Evaluating roots of polynomials 225

38. Equations of Second, Third and Fourth Degree 225
39. Bounds of Roots 232
40. Sturm’s Theorem 238
41. Other Theorems on the Number of Real Roots 244
42. Approximation of Roots 250

Chapter 10.
Fields and polynomials 257

43. Number Rings and Fields 257
44. Rings 260
45. Fields 267
46. Isomorphisms of Rings (Fields). The Uniqueness of the Field of Complex Numbers 272
47. Linear Algebra and the Algebra of Polynomials over an Arbitrary Field 276
48. Factorization of Polynomials into Irreducible Factors 281
49. Theorem on the Existence of a Root 290
50. The Field of Rational Fractions 297

Chapter 11.
Polynomials in several unknowns 303

51. The Ring of Polynomials in Several Unknowns 303
52. Symmetric Polynomials. 312
53. Symmetric Polynomials Continued 319
54. Resultant. Elimination of Unknown. Discriminant 329
55. Alternative Proof of the Fundamental Theorem of the Algebra of
Complex Numbers 337

Chapter 12.
Polynomials with rational coefficients 341

56. Reducibility of Polynomials over the Field of Rationals 341
57. Rational Roots of Integral Polynomials 345
58. Algebraic Numbers 349

Chapter 13.
Normal form of a matrix 355

59. Equivalence of $\lambda$-matrices.  355
60. Unimodular $\lambda$-matrices. Relationship Between Similarity of
Numerical Matrices and the Equivalence of their Characteristic Matrices 362
61. JordanNormalForm  370
62. MinimalPolynomials  377

Chapter 14.
Groups. 382

63. Definition of a Group. Examples 382
64. Subgroups 388
65. Normal Divisors, Factor Groups, Homomorphisms 394
66. Direct Sums of Abelian Groups 399
67. Finite Abelian Groups 406

Bibliography 414
Index 416

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### 12 Responses to Higher Algebra – Kurosh

1. Siddharth says:

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2. Devarsh says:

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• rd says:

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3. MirCollector says:

I will get this book in a week’s time.

4. thank you fir your hard work

5. galib20 says: