In this post we will see Problems in Higher Algebra – *Faddeev, Sominsky *

This book of problems in higher algebra grew out of a course

of instruction at the Leningrad State University and the Herzen

Pedagogical Institute. It is designed for students of universities

and teacher’s colleges as a problem book in higher algebra.The problems included here are of two radically different types. On

the one hand, there are a large number of numerical examples aimed at developing computational skills and illustrating the basic propositions of the theory. The authors believe that the number of

problems is sufficient to cover work in class, at home and for tests.On the other hand, there are a rather large number of problems

of medium difficulty and many which will demand all the initiative and ingenuity of the student. Many of the problems of

this category are accompanied by hints and suggestions to be

found in Part II. These problems are starred.

Answers are given to all problems, some of the problems are

supplied with detailed solutions.

This book was translated from the Russian by* George Yankovsky*. The book was published by first Mir Publishers in 1978.

All credits to the *original uploader.*

PDF | 3.4 MB | Pages: 318 | Bookmarked | OCR | Cover

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Table of Contents

Contents

**Part I**

PROBLEMS

**CHAPTER I. COMPLEX NUMBERS 11**

1. Operations on Complex Numbers 11

2. Complex Numbers in Trigonometric Form 13

3. Equations of Third and Fourth Degree 19

4. Roots of Unity 21

**CHAPTER 2. EVALUATION OF DETERMINANTS 25**

1. Determinants of Second and Third Order 25

2. Permutations 26

3. Definition of a Determinant 27

4. Basic Properties of Determinants 29

5. Computing Determinants 31

6. Multiplication of Determinants 51

7. Miscellaneous Problems 56

**CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS 61**

1. Cramer’s Theorem 61

2. Rank of a Matrix 64

3. Systems of Linear Forms 66

4. Systems of Linear Equations 68

**CHAPTER 4. MATRICES 76**

1. Operations on Square Matrices 76

2. Rectangular Matrices. Some Inequalities 83

**CHAPTER 5. POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 88**

1. Operations on Polynomials. Taylor’s Formula. Multiple Roots 88

2. Proof of the Fundamental Theorem of Higher Algebra and Allied

Questions 92

3. Factorization into Linear Factors. Factorization into Irreducible

Factors in the Field of Reals. Relationships Between Coefficients

and Roots 93

4. Euclid’s Algorithm 97

5. The Interpolation Problem and Fractional Rational Functions 100

6. Rational Roots of Polynomials. Reducibility and Irreducibility over

the Field of Rationals 103

7. Bounds of the Roots of a Polynomial 107

8. Sturm’s Theorem 108

9. Theorems on the Distribution of Roots of a Polynomial 111

10. Approximating Roots of a Polynomial 115

**CHAPTER 6. SYMMETRIC FUNCTIONS 116**

1. Expressing Symmetric Functions in Terms of Elementary Symmetric

Functions. Computing Symmetric Functions of the Roots of an

Algebraic Equation 116

2. Power Sums 121

3. Transformation of Equations 123

4. Resultant and Discriminant 124

5. The Tschirnhausen Transformation and Rationalization of the

Denominator 129

6. Polynomials that Remain Unchanged under Even Permutations of the Variables. Polynomials that Remain Unchanged under Circular

Permutations of the Variables 130

**CHAPTER 7. LINEAR ALGEBRA 133**

1. Subspaces and Linear Manifolds. Transformation of Coordinates 133

2. Elementary Geometry of n-Dimensional Euclidean Space 135

3. Eigenvalues and Eigenvectors of a Matrix 139

4. Quadratic Forms and Symmetric Matrices 141

5. Linear Transformations. Jordan Canonical Form 146

**PART II**

**HINTS TO SOLUTIONS**

CHAPTER 1. COMPLEX NUMBERS 151

CHAPTER 2. EVALUATION OF DETERMINANTS 153

CHAPTER 4. MATRICES 159

CHAPTER 5. POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 160

CHAPTER 6. SYMMETRIC FUNCTIONS 164

CHAPTER 7. LINEAR ALGEBRA 166

**PART III**

ANSWERS AND SOLUTIONS

CHAPTER 1. COMPLEX NUMBERS 168

CHAPTER 2. EVALUATION OF DETERMINANTS 186

CHAPTER 3. SYSTEMS OF LINEAR EQUATIONS 196

CHAPTER 4. MATRICES 203

CHAPTER 5. POLYNOMIALS AND RATIONAL FUNCTIONS OF ONE VARIABLE 221

CHAPTER 6. SYMMETRIC FUNCTIONS 261

CHAPTER 7. LINEAR ALGEBRA 286

INDEX 313

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