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Beaumont R.A., Pierce R.S. — The Algebraic Foundations of Mathematics |
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Предметный указатель |
Abscissa 299
Absolute value 132 292
Addition in a ring 107
Addition of complex numbers 287 301
Addition of integers 101
Addition of matrices 430
Addition of natural numbers 89
Addition of polynomials 314
Addition of rational numbers 219
Addition of real numbers 241
Algebraic number 385
Amplitude 304
Archimedes 226
Archimedes’ Principle 233
Argument 304
Associate 331
Associative law for addition 90
Associative law for matrix multiplication 437
Associative law for multiplication 93
Associative law for set operations 37
Base 138
Basis of induction 57 68
Binary operation for rings 107 114
Binary operation for sets 30
Binary system 140
Binomial coefficients 61 64
Binomial theorem 63
Bombelli, Rafael 286
Bounds for roots 374
C in a ring 121
C of multiplication 93 96
Cancellation law of addition 90 96
Cantor, Georg 18
Cantor’s theorem 279
Cardan, Girolamo 309
Cardan’s formulas 364
Cardinal number of a finite set 18 86
Cardinal number of a set 20 83
Cardinality 18 20 166
Cartesian coordinates 299
Casting out nines 179
Characteristic of an integral, domain 210
Chinese remainder theorem 188
Coefficient 313
Column matrix 429
Common divisor 144 150
Common multiple 151
Commutative law for addition 90
Commutative law for multiplication 93
Commutative ring 108
Comparison Test 269
Complement of a set 31
Complete factorization 336
Complete ordered integral, domain 251
Completing the square 296
Complex conjugate 291
complex numbers 2 287
Complex plane 300
Composite number 153
Congruence 176
Congruence classes 182
Congruent modulo m 176 215
conjugate 291
Consistent systems of, equations 411
Constant polynomial 317
Constant term 317
Contrapositive 7
Convergent sequence 258
Convergent Series 264
CONVERSE 7
Coordinate line 230
Coordinate plane 299
Coordinate system 230 238
Course of values induction 68
Cubic equation 361
Decimal fraction 226
Decimal representation of real numbers 277
Decimal representation, of, rational numbers 282
Dedekind cuts 237
Dedekind, Richard 224
Definition 7
Degree of a polynomial 321 397
Degree of an algebraic number 386
DeMoivre’s Theorem 305
Density property 233
Denumerable 23
Derivative 338
Descartes, Rene 299
Determinant 412 429
Diagonal method 280
Difference of natural numbers 96
Difference of sets 36
Dimensions of a matrix 429
Diophantine equation 169
Direct proof 9
Discriminant 368
Disjoint sets 44
Disjunctive normal form, theorem 40
Distance 299
Distributive law for natural, numbers 93
Distributive law for set operations 39
Divergent series 264
Division algorithm for integers 135
Division algorithm for polynomials 322
Division in an integral domain 122
Division of polynomials 326
Divisor of zero 122
Domain of polynomials 317 394
Duodecimal system 139
E of a system of equations 416
Echelon form of a matrix 447
Element of a matrix 429
Element of a set 11 17
Elementary symmetric, polynomial 406
Elementary transformation, matrix 451
Elementary transformations of a system of equations 413
Elementary transformations, of, a matrix 446
Empty set 13 17
Equal, matrices 430
Equal, polynomials 313
Equal, sets 11
Equivalence class 216
Equivalence of sets 20
Equivalence of statements 6
Equivalence relation 214
Equivalent systems of, equations 412
Euclid 148 160
Euclidean algorithm for polynomials 330
Euclidean algorithm, for integers 148
Euler, Leonhard 189
Euler’s theorem 191
EXPONENT 194 206
Factor 122 326
Factor Theorem 347
Fermat conjecture 172
Fermat numbers 162
Fermat, Pierre 162
Fermat’s theorem 191
Fibonacci sequence 79 151 152
Field 204
Finite ordinal numbers 84
Finite sets 18 20 86
Fundamental Theorem of Arithmetic 155
Fundamental theorem of decimal representation 272
Fundamental theorem of symmetric polynomials 407 466
Fundamental theorem, of, algebra 355 474
General associative law 115 117
General commutative law 115 117
General distributive law 118
Godel numbering 165
| Graph 369
Greatest common divisor for integers 146 148 150 157
Greatest common divisor for polynomials 327 332
Greatest element 130
Greatest integer function 278
Greatest lower bound 249
Homogeneous system of, equations 424
Identity 4
Identity element 93 121
Identity matrix 443
Imaginary part of a complex, number 291
Implication 4
Incongruent solutions, modulo m 183
Inconsistent system of, equations 411
Indeterminate 317
Index of summation 115
Index set 37
Indirect proof 9
Induction hypothesis 57 68
Induction step 57 68
Inductive definitions 79
Inequality 129
Infinite decimal sequence 228
Infinite sequence 258
Infinite series 264
Infinite set 18 20
Integers 2 100
Integral domain 121
Intersection of sets 30 36
Inverse of a square matrix 443
Inverse of an element 205
Inverse of an implication 7
Irrational number 224
Irreducible polynomial 333 357 358
Isomorphic rings 111
Isomorphism 111
largest element 130
Latin square 185
Law of substitution 8
Leading coefficient 321
Least common multiple of polynomials 332
Least common multiple, of, integers 151 157
Least element 130
Least upper bound 249
LIMIT 260
Linear congruence 181
Lower bound 248
Lower bound for roots 374
Mathematical induction 53
Matrix 428
Matrix of coefficients 434
Maximum element 130
Measure of a set 41 45
Mersenne number 162
Method of infinite descent 174
Method of infinite descent, m-fold root 348
Minimum element 130
Minimum polynomial 386
Modulus 176 292
Monic associate 328
Monic polynomial 328
Monic polynomial, mth root 256
Multiple factor 345
Multiple root 349
Multiplication in a ring 107
Multiplication of complex, numbers 287 305
Multiplication of integers 104
Multiplication of matrices 433
Multiplication of natural numbers 91 92
Multiplication of polynomials 315
Multiplication of rational numbers 219
Multiplication of real numbers 244
Multiplicity 348
Natural numbers 2 82 85
Negation in a ring 107
Negation of complex numbers 287
Negation of integers 101
Negation of polynomials 315
Negation of rational numbers 219
Negation of real numbers 243
Negative elements 127
negative numbers 100 244
Negative of a matrix 432
Negative of a polynomial 315
NIM 140
Nonnegative elements 128
Nonnegative real numbers 244
Nonsingular matrix 443
Nonsingular matrix, n-place decimal approximation 273
Nonsingular matrix, n-place decimal fraction 270
Nonsingular matrix, n-rowed square matrix 439
Nonsingular matrix, nth roots of unity 308
Number of divisors 156
One-to-one correspondence 19
Order modulo m 194 352
Ordered field 207
Ordered integral domain 126
Ordered pair 24
Ordering of integers 125
Ordering of natural numbers 95
Ordering of rational numbers 219
Ordering of real numbers 240
Ordinate 299
Origin 230 298
Orthogonal latin square 185
Pairwise disjoint collection of, sets 44
Parallelogram rule 302
Partial sum 264
Partially ordered set 248
Partition 218
Pascal triangle 61
Peano’s axioms 87
Perfect number 163
Permutation 20
Polar representation 303
Polynomial 313
Polynomial in several indeterminates 394
Positive elements 127
Positive integers 128
Positive real numbers 244
Power set 26
Prime characteristic 210
Prime number 68 153 159
Prime number theorem 161
Prime pair 161
Primitive root 195 352
Principle of mathematical, induction 57 76
Probability measure 43
Product of matrices 433
Product of natural numbers 91 92
Product of polynomials 315
Product of sets 25
Product sign 118
Proofs 8
Proper divisor of zero 122
Proper subset 15
Pythagoras’ theorem 224 234
Quadratic equation 296
Quartic equation 365
Quotient 124 136 325
Range of a variable 3
Rational numbers 2 200 218
Real numbers 2 224 238
Real part of a complex number 291
recursive definitions 79
Reduced cubic equation 362
Reducible polynomial 333
Reflexive law 215
Relation 213
Relatively prime integers 147
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