Higher algebra

A. Kurosh

За особеностите на екземпляра

Състояние:: Много добро
Забележка:: Леки пожълтявания по хартията.
Език:: Английски
Наличност:: 1 бр.

За изданието

Издателство:: Mir
Град на издаване:: Moscow
Година:: 1988 г.
Език:: Английски
Страници:: 428
Корици:: Твърди
Категория:: Математика
Формат:: 215/145

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. 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