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

the equations by means of radicals, even radicals within radicals. It

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

down by means of radicals.

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.

All credits to the *original uploader.*

You can get the book here.

Update 3 June 2021: added The Internet Archive Link

Table of Contents

**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.**

** Quadratic forms 161**

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**

Desperadomar,

Is this blog owned jointly by you and damitr ?

LikeLike

Sir, can you please upload these book on 4 shared

LikeLike

Devarsh ,please wait a while ,the links will up soon on 4shared.

LikeLike

links still dead…

LikeLike

rd, The 4shared link is working fine. I checked it now.LikeLike

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

LikeLike

Pingback: Algebra and Number Theory | Mathematica

thank you fir your hard work

LikeLike

link not working archive

LikeLike

@gailb20, The 4shared link is still working.

LikeLike

Thank you.

LikeLike

4sharelink: on clicking Download ;it is an endless loop.

LikeLike

The link is not available. Can you please upload this to Internet archive.

LikeLike

Give me a couple of days, I am in process of updating all the dead links.

LikeLike