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| Birkhoff G., Mac Lane S. — A Survey of Modern Algebra |
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| Предметный указатель |
Abelian group 133
Absolute value 10
Absolute value of complex number 110
Absorption law 362
Addition see also "Sum"
Addition of cardinal numbers 390
Addition of complex numbers 107
Addition of inequalities 10
Addition of matrices 220ff.
Addition of polynomials 62ff.
Addition of vectors 170
Additive group 170 269 415 446
Additive inverse 2
Adjoint of matrix 325
Affine geometry 305ff.
Affine group 269
Affine independence 309
Affine space 305
Affine subspace 306
Affine transformation 270
algebraic 421
Algebraic extension 422
Algebraic function field 427
Algebraic integer 443
Algebraic number 435
Algebraic number field 438
Algebraic variety 410
Algebraically closed 437
Algebraically complete 437 454
Algebraically independent 73
Alias-alibi 271
Alternating group 154
Angle between vectors 202
Annihilator 211
Anti-automorphism 232
Anti-symmetric law 357 372
Archimedean property 99
Argument of complex number 110
Arithmetic, fundamental theorem of 23
Associate 76
Associative law 2
Associative law for groups 127 133
Associative law for matrices 224
Associative law for rings 2
Associative law for transformations 127
Associative law, general 13
atom 378
Augmented matrix 247
Automorphism 36
Automorphism of a group 157
Automorphism of a vector space 235
Automorphism, inner 158
Automorphism, involutory 232
Automorphism, outer 158
Axiom of Choice 129 384
Barycentric coordinates 310
Base of parallelepiped 328
Basis 176 193
Basis for ideals 402 411
Basis of vector spaces 176 193
Basis, change of 260
Basis, normal orthogonal 203ff.
Basis, normal unitary 302
Bijection 33 129
Bilinear form 281
Bilinear function 251
Bilinear multiplication 225
Bilinearity 199
Binary operation 33
Binary relation 34
Binomial coefficients 14
Block multiplication 234
Boolean algebra 359 361
Boolean algebra, free 377
Boolean algebra, isomorphism of 377
Boolean function 368
Boolean function polynomial 368
Boolean function, minimal polynomial 370
Bounds (lower, upper, etc.) 96
Cancellation law 3 391
Cancellation law for cardinal numbers 391
Cancellation law for groups 134
Canonical form 187 248 277
Canonical form for linear forms 281
Canonical form for matrices 187
Canonical form for quadratic forms 283ff.
Canonical form, bilinear 282
Canonical form, disjunctive 369
Canonical form, Jordan 355
Canonical form, primary rational 353
Canonical projection 207
Cantor's diagonal process 386 436
cardinal number 381 387
Cartesian product 34
Cayley — Hamilton theorem 340
Cayley's theorem 139 269 398 482
Center of a group 144
Centroids 308
Chain 376
Characteristic 415ff.
Characteristic equation 333
Characteristic polynomial 331ff.
Characteristic root 265 332
Characteristic vector 265 332
class see "Set"
Closure under an operation 1
Codomain of transformation 33 241
Cofactor 319
Column equivalent 249
Column operation 248
Column rank 250
Column vector 232
Commutative group 133
Commutative law 2
Commutative law, general 14
Commutative ring 1 69
Commutator 163
Companion matrix 338
Complement in a lattice 375
Complementarity law 359
Complementary subspaces 196
Complementation 362
Complete ordered domain 98
Complete ordered field 98
Complete set of invariants 277
Completing the square 119
Complex number 41 107ff.
Complex plane 110
Composite 127
Congruence 25 43 see
Congruence relation 164
Congruent 25 43 405
Congruent matrices 285
conic 283 296 316
Conjugate algebraic numbers 448
Conjugate complex numbers 117
Conjugate diameter 308
Conjugate in a group 158
Conjugate quaternions 257
Conjugate subfields 474
Conjugate subgroups 163
Conjugate vector space 210
Consistency principle 359
Constructive proof 92
Contain (a set) 358
Content of polynomial 86
Continuum, cardinal number of 387
Coordinates of vectors 193 209 262
Coordinates, change of 260ff.
Coordinates, homogeneous 313
| Coset 146 207
Countable set 384 436
Cover (in partial order) 372
Cramer's rule 325
Critical points 291
Cubic equation 119
Cubic equation, irreducible case 474
Cubic equation, trigonometric solution 102
Cut, Dedekind 104
CYCLE 150
Cyclic group 140
Cyclic permutation 150
Cyclic subspace 344
Cyclotomic equation 112
Cyclotomic polynomial 89
De Moivre formulas 111
Decimals 97
Decomposition of ideal 450
Decomposition of matrices, linear transformations 346ff.
Dedekind cut (axiom) 104
Defining relation 142
Degree 62 66
Degree of extension field 429
Degree of polynomial 62
Denumerable set 384
Dependence, linear 176
Derivative, formal 465
Determinant 48 318ff.
Determinant, rank 326
Diagonal matrix 228
Diagonal process 386 436
Diagonally dominant 323
Diagram, for partial order 372
Dihedral groups 143
Dilemma, Pythagorean 94
Dimension of vector space 169 178
Direct product 156
Direct sum 195
Direct sum of matrices 347
Direct sum of rings 396
Direct sum of subspaces 195 346
Discriminant 117
Discriminant of cubic 120
Disjunctive canonical form 369
Distance 201
Distributive lattice 363
Distributive law 2 134 170
Distributive law for Boolean algebra 362
Distributive law for cardinal numbers 390
Distributive law for matrices 224
Distributive law for sets 359
Distributive law, general 14
Distributive law, right 4
Divisible 16
Division algebra 396 414
Division algorithm 18 74 440
Division ring 256 401
Divisor 16
Divisor of zero see "Zero divisor"
Divisor, greatest common 19
Domain 33 241
Domain, integral 3
Domain, ordered 9 98
Domain, unique factorization 84
Dual basis 210
Dual isomorphism 473
Dual numbers 397
Dual space 210
Duality principle 212 373
Dualization law 359
Duplication of cube 434
Echelon matrix 184 185
Eigenvalue 265 332
Eigenvector 265
Eisenstein's irreducibility criterion 88
Elementary column operation 248
Elementary divisors 353
Elementary matrix 243ff. 321
Elementary row operation 181
Elementary symmetric polynomial 154
Elimination 48 180
Elliptic function field 427
Epimorphic image 400
Epimorphism 399
Equality of functions 127
Equality of transformations 127
Equality, laws of 3 26
Equations see also "Polynomial"
Equations of stable type 122
Equations, simultaneous linear 47
EQUIVALENCE 248 277
Equivalence relation 34 164
Equivalence, under a group 277ff.
Erlanger Program 132
Euclidean algorithm 18
Euclidean algorithm for Gaussian integers 441
Euclidean algorithm for polynomials 81
Euclidean group 131 276
Euclidean vector space 200ff.
Even permutation 153
Exponentiation of cardinal numbers 392
Exponents, laws of 14
Extension 45
Extension field 421
Extension field, iterated 431ff.
Factor 16
Factor group see "Quotient-group"
Fermat theorem 28 418
Field 38 133
Field of quotients 43
Field of sets 378
Finite dimensional 178
Finite extension 429
Finite field 456ff.
Finite induction 13
Finite induction, second principle of 15
Finite set 383
Formal derivative 465
Four group 151
Full linear group 268
Fully reducible 347
Function 32
Fundamental theorem of algebra 113ff.
Fundamental Theorem of Arithmetic 23
Fundamental Theorem of Galois Theory 472
Fundamental theorem of ideal theory 450
Fundamental theorem on symmetric polynomials 154
Galileo paradox 384
Galois field 458
Galois group 459ff.
Gauss elimination 48 180
Gauss' lemma 85
Gaussian domain see "Unique factorization domain"
Gaussian integer 439
Generation of subalgebras 369
Generators of a field extension 421
Generators of group 141
Generators of vector space see "Span"
Gram-Schmidt process 204
Greatest common divisor 19 81 407
Greatest lower bound 96 374
Group 124ff. 133
Group of transformations 126 130
Group, abstract 133
Group, algebra 397
Group, cyclic 140
Hadamard determinant theorem 331
Hermitian form 302
Hermitian matrix 300ff.
Hilbert basis theorem 412
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