Problems in Higher Algebra

PREFACE
TABLE OF CONTENTS
CHAPTER I - PROBLEMS (COMPLEX NUMBERS)
	1. RECKONING WITH COMPLEX NUMBERS
	2. COMPLEX NUMBERS IN POLAR FORM
	3. EQUATIONS OF THE THIRD AND FOURTH DEGREE
	4. ROOTS OF UNITY
CHAPTER II - PROBLEMS (COMPUTATION OF DETERMINANTS)
	1. SECOND AND THIRD ORDER DETERMINANTS
	2. PERMUTATIONS
	3. DEFINITION OF DETERMINANTS
	4. FUNDAMENTAL PROPERTIES OF DETERMINANTS
	5. EVALUATION OF DETERMINANTS
	6. MULTIPLICATION OF DETERMINANTS
	7. MISCELIANEOUS PROBLEMS
CHAPTER III - PROBLEMS (SYSTEMS OF LINEAR EQUATIONS)
	1. CRAMER\'S RULE
	2. THE RANK OF A MATRIX
	3. SYSTEMS OF LINEAR FORMS
	4. SYSTEMS OF LINEAR EQUATIONS
CHAPTER IV - PROBLEMS (MATRICES)
	1. OPERATIONS WITH SQUARE MATRICES
	2. RECTANGULAR MATRICES. CERTAIN INEQUALITIES
CHAPTER V - PROBLEMS (POLYNOMIALS AND RATIONAL FUNCTIONS OF A SINGLE INDETERMINATE)
	1. PROPERTIES OF POLYNOMIALS. TAYLOR IS FORMULA, MULTIPLE ROOTS
	2. PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA AND RELATED QUESTIONS
	3. FACTORING INTO LINEAR FACTORS. FACTORING INTO FACTORS THAT ARE IRREDUCIBLE IN THE FIELD OF REAL NUMBERS. FORMULAS CONNECTING COEFFICIENTS AND ROOTS.
	4. EUCLIDEAN ALGORITHM
	5. THE INTERPOLATION PROBLEM; RATIONAL FUNCTIONS
	6. RATIONAL ROOTS OF POLYNOMIALS REDUCIBILITY AND IRREDUCIBILITY IN THE FIELD OF RATIONAL NUMBERS
	7. BOUNDS FOR THE ROOTS OF POLYNOMIALS
	8. STURM\'S THEOREM
	9. MISCELLANEOUS METHODS FOR THE LOCATION OF ROOTS OF POLYNOMIALS
	10. APPROXIMATE CALCULATION OF THE ROOTS OF POLYNOMIALS
CHAPTER VI - PROBLEMS (SYMMETRIC FUNCTIONS)
	1. SYMMETRIC FUNCTIONS IN TERMS OF THE ELEMENTARY SYMMETRIC FUNCTIONS. CALCULATION OF THE SYMMETRIC FUNCTIONS OF THE ROOTS OF AN ALGEBRAIC ~UATION.
	2. POWER SUMS
	3. TRANSFORMATION OF EQUATIONS
	4. RESULTANTS AND DISCRIMINANTS
	5. TSCHIRNHAUS TMNSFORMATION AND THE ELIMINATION OF IRRATIONALITIES IN THE DENOMINATOR.
	6. POLYNOMIALS THAT ARE INVARIANT UNDER EVEN PERMUTATION OF THE VARIABLES. POLYNOMIALS THAT ARE INVARIANT UNDER CIRCULAR PERMUTATION OF THE VARIABLES.
CHAPTER VII - PROBLEMS (LINEAR ALGEBRA)
	1. SUBSPACES AND LINEAR VARIETIES. COORDINATE SUBSPACES.
	2. ELEMENTARY GEOMETRY OF n-DIMENSIONAL EUCLIDEAN SPACE.
	3. PROPER VALUES AND PROPER VECTORS OF A MATRIX.
	4. QUADRATIC FORMS AND SYMMETRIC MATRICES
	5. LINEAR SPACES. THE JORDAN CANONICAL FORM.
CHAPTER I - HINTS (COMPLEX NUMBERS)
50
100
CHAPTER II - HINTS (COMPUTATION OF DETERMINANTS)
150
200
250
300
CHAPTER IV - HINTS (MATRICES)
CHAPTER V - HINTS (POLYNOMIALS AND RATIONAL FUNCTIONS IN A SINGLE INDETERMINATE)
600
650
700
CHAPTER VI - HINTS (SYMMETRIC FUNCTIONS)
800
850
CHAPTER VII - HINTS (LINEAR ALGEBRA)
900
950
CHAPTER I - SOLUTIONS (COMPLEX NUMBERS)
	10
	20
	30
	40
	50
	60
	70
	80
	90
	100
	110
	120
	130
CHAPTER II - SOLUTIONS (COMPUTATION OF DETERMINANTS)
	140
	150-160
	170-200
	210
	220
	230-240
	250
	260
	270-280
	290
	300
	310
	320
	330
	340-350
CHAPTER III - SOLUTIONS (SYSTEMS OF LINEAR EQUATIONS)
	360-370
	380-400
	410-420
	430
	440
	450
	460
CHAPTER IV - (SOLUTIONS MATRICES)
	470
	480
	490
	500
	510
	520
	530
CHAPTER V - SOLUTIONS (POLYNOMIALS AND RATIONAL FUNCTIONS OF A SINGLE INDETERMINATE)
	540
	550
	560
	570
	580
	590
	600
	610
	620
	630
	640
	650
	660
	670
	680
	690
	700
	710
	720
	730
	740
	750
CHAPTER VI - SOLUTIONS (SYMMETRIC FUNCTIONS)
	760
	770
	780
	790
	800
	810
	820
	830
	840
	850
	860
	870
CHAPTER VII - SOLUTIONS (LINEAR ALGEBRA)
	880
	890
	900
	910
	920
	930
	940
	950
	960
	970
	980