Problems in Higher Algebra – Faddeev, Sominsky

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.

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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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2 Responses to Problems in Higher Algebra – Faddeev, Sominsky

  1. m95 says:

    thanks a lot, great effort

  2. Prithviraj says:

    thanks!!!!!!!!!!

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