Subject Index
Generally, non-X, un-X, in-X is listed under X.
abelian group 25 augmentation ideal 132, 167, 293
abelianization 67 augmented algebra 132
absolute value 273, 312 automorphism 27, 80, 211
absorptive law 53 axiom of choice 10,60
acquaintanceship graph 16
acyclic 17 Baer's criterion 116
addition (mod 2) 81 balanced mapping 122
additive function 173 basis 105, 398
- functor II 0 - theorem for abelian groups 38
adjacency matrix 23 bidual 126
adjoint associativity 119, 123 bifunctor 87
affine group 227, 244 bilinear form 249ff.
aleph 2 - mapping 117, 122
algebra 131 bimodule 83
algebraic element 192 binary form 249
- equation 376 Binet-Cauchy identity 182
- extension 193, 403 Boolean algebra 70, 134
- integer 134 - polynomial 71
- set 378 - ring 80
algebraically closed 201, 285, 431 Brauer group 152, 296
- dependent 402 Burnside problem 28, 178
alternating form 256, 298, 301
- matrix 298 cancellation 31
- group 33 cardinal (number) Iff.
anisotropic part 257, 270, 293 Cartan-Dieudonne theorem 258
annihilator 84 Castelnuovo-Zariski theorem 408
anti-chain 8, 63f. casus irreducibilis 247
anticommutative 166f. category 65
antiderivation 180 Cauchy sequence 275, 314
antihomomorphism 66, 83 Cayley's theorem 32, 135, 214
approximation theorem 316, 369 central chain 39
archimedean absolute value 313 - simple algebra 150
- ordering 277 centralizer 30, 149
Arf invariant 303 centrally primitive idempotent 103
arrow 17 centre 30, 131
Artin's theorem 209 chain 8
Artin-Schreier extension 444 - condition 60
Artin-Schreier theory 279ff. character (group) 125
Artinian module, ring 89 characteristic of field 189
assassinator 380 - - prime ideal 297
associated elements 349 - - ring 80, 104
- prime ideal 380 - function 6
associative law 25, 53 - polynomial 153, 175
atom 75, 193, 349 - subgroup 46
atomic Boolean algebra 78 Chevalley's lemma 333
- domain 350 chief factor, series 36
459
, 460 Basic Algebra
Chinese remainder theorem 102, 115 - - rational function 405
class 65 - - symmetric group 26
- equation 30 Delian problem 194
- of nilpotence 40 denominator 335, 354f.
classical logic 71
Clifford algebra 260 dense embedding 275, 393
- group 265f. - functor 67
closed set 378ff. density principle 228
cofinal 14 dependence relation 397ff.
cofinite subset 70 derivation 171, 415ff.
1. S. Cohen's theorem 395 derivative 204
coimage, cokernel 85 derived group, series 39
comaximal ideals 102 determinant 182
comma category 69 - of a form 252
commutative diagram 85 diagonal argument 8
-law 25,53 diamond lattice 57
- ring 79 dicyclic group 50
commutator 39, 42 different 395
companion matrix 155 digraph 17
complement 56, 92 dihedral group 26, 50
complementary graph 16 Dilworth's theorem 18
complete lattice 55 dimension 140, 249, 387, 403
- ordered field 275 - index 295
- space 314 direct power 87
completely primary ring 104, 187, 386 - product 27, 40, 87
completion 277, 315 - - of rings 101
composite of fields 428 - sum of modules 87
composition series 36 directed graph 52
compound matrix 182 - system 64, 202
concrete category 66 Dirichlet box principle 2
conductor 391 Dirichlet's theorem 222
cone 279f. discrete absolute value 313
congruent matrices 250 - rank 1 valuation 308
conjugacy class 30 discriminant 231, 263
conjugate 30, 42, 199,213 disjunctive normal form 72
conjunctive normal form 72 distributive lattice 59, 96
connected graph 19 - law 59, 79, 119
conorm mapping 370 divisible module 116
consistent system 377 division algebra 133, 150
constant 171,415 - ring 80
continuum hypothesis 7 dominate 332
contracted ideal 356 dual basis lemma 114
converge 275 - categories 67
co(ntra)variant functor 66 - group 125
coordinate ring 377 - homomorphism 71
core of a field 280 duality 67f.
coset (space) 27ff.
countable 2 edge IS
cubic equation 238, 245 Einseinheiten 326
cubical norm 318 Eisenstein polynomial 345
cycle notation 33 Eisenstein's criterion 200
cyclic extension 235 elementary abelian group 26
- group 27 embedded component 385
- module 82 endomorphism 27, 80
cyclotomic polynomial 219ff. - ring 82, 135
endpoint 15
De Morgan's laws 70 enumerable 2
decomposition lemma 63, 361, 365 equipotent 1
Dedekind domain 115, 365ff., 390 equivalence xii, 67
Dedekind's lemma 206, 231 equivalent valuations 311
defining relation 26 - absolute values 315
degree of field extension 190 essential extension 129
- - polynomial 166 Euclid's Elements 193
Generally, non-X, un-X, in-X is listed under X.
abelian group 25 augmentation ideal 132, 167, 293
abelianization 67 augmented algebra 132
absolute value 273, 312 automorphism 27, 80, 211
absorptive law 53 axiom of choice 10,60
acquaintanceship graph 16
acyclic 17 Baer's criterion 116
addition (mod 2) 81 balanced mapping 122
additive function 173 basis 105, 398
- functor II 0 - theorem for abelian groups 38
adjacency matrix 23 bidual 126
adjoint associativity 119, 123 bifunctor 87
affine group 227, 244 bilinear form 249ff.
aleph 2 - mapping 117, 122
algebra 131 bimodule 83
algebraic element 192 binary form 249
- equation 376 Binet-Cauchy identity 182
- extension 193, 403 Boolean algebra 70, 134
- integer 134 - polynomial 71
- set 378 - ring 80
algebraically closed 201, 285, 431 Brauer group 152, 296
- dependent 402 Burnside problem 28, 178
alternating form 256, 298, 301
- matrix 298 cancellation 31
- group 33 cardinal (number) Iff.
anisotropic part 257, 270, 293 Cartan-Dieudonne theorem 258
annihilator 84 Castelnuovo-Zariski theorem 408
anti-chain 8, 63f. casus irreducibilis 247
anticommutative 166f. category 65
antiderivation 180 Cauchy sequence 275, 314
antihomomorphism 66, 83 Cayley's theorem 32, 135, 214
approximation theorem 316, 369 central chain 39
archimedean absolute value 313 - simple algebra 150
- ordering 277 centralizer 30, 149
Arf invariant 303 centrally primitive idempotent 103
arrow 17 centre 30, 131
Artin's theorem 209 chain 8
Artin-Schreier extension 444 - condition 60
Artin-Schreier theory 279ff. character (group) 125
Artinian module, ring 89 characteristic of field 189
assassinator 380 - - prime ideal 297
associated elements 349 - - ring 80, 104
- prime ideal 380 - function 6
associative law 25, 53 - polynomial 153, 175
atom 75, 193, 349 - subgroup 46
atomic Boolean algebra 78 Chevalley's lemma 333
- domain 350 chief factor, series 36
459
, 460 Basic Algebra
Chinese remainder theorem 102, 115 - - rational function 405
class 65 - - symmetric group 26
- equation 30 Delian problem 194
- of nilpotence 40 denominator 335, 354f.
classical logic 71
Clifford algebra 260 dense embedding 275, 393
- group 265f. - functor 67
closed set 378ff. density principle 228
cofinal 14 dependence relation 397ff.
cofinite subset 70 derivation 171, 415ff.
1. S. Cohen's theorem 395 derivative 204
coimage, cokernel 85 derived group, series 39
comaximal ideals 102 determinant 182
comma category 69 - of a form 252
commutative diagram 85 diagonal argument 8
-law 25,53 diamond lattice 57
- ring 79 dicyclic group 50
commutator 39, 42 different 395
companion matrix 155 digraph 17
complement 56, 92 dihedral group 26, 50
complementary graph 16 Dilworth's theorem 18
complete lattice 55 dimension 140, 249, 387, 403
- ordered field 275 - index 295
- space 314 direct power 87
completely primary ring 104, 187, 386 - product 27, 40, 87
completion 277, 315 - - of rings 101
composite of fields 428 - sum of modules 87
composition series 36 directed graph 52
compound matrix 182 - system 64, 202
concrete category 66 Dirichlet box principle 2
conductor 391 Dirichlet's theorem 222
cone 279f. discrete absolute value 313
congruent matrices 250 - rank 1 valuation 308
conjugacy class 30 discriminant 231, 263
conjugate 30, 42, 199,213 disjunctive normal form 72
conjunctive normal form 72 distributive lattice 59, 96
connected graph 19 - law 59, 79, 119
conorm mapping 370 divisible module 116
consistent system 377 division algebra 133, 150
constant 171,415 - ring 80
continuum hypothesis 7 dominate 332
contracted ideal 356 dual basis lemma 114
converge 275 - categories 67
co(ntra)variant functor 66 - group 125
coordinate ring 377 - homomorphism 71
core of a field 280 duality 67f.
coset (space) 27ff.
countable 2 edge IS
cubic equation 238, 245 Einseinheiten 326
cubical norm 318 Eisenstein polynomial 345
cycle notation 33 Eisenstein's criterion 200
cyclic extension 235 elementary abelian group 26
- group 27 embedded component 385
- module 82 endomorphism 27, 80
cyclotomic polynomial 219ff. - ring 82, 135
endpoint 15
De Morgan's laws 70 enumerable 2
decomposition lemma 63, 361, 365 equipotent 1
Dedekind domain 115, 365ff., 390 equivalence xii, 67
Dedekind's lemma 206, 231 equivalent valuations 311
defining relation 26 - absolute values 315
degree of field extension 190 essential extension 129
- - polynomial 166 Euclid's Elements 193
