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Li H., Gras G. — Class Field Theory: From Theory to Practice
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Название: Class Field Theory: From Theory to PracticeАвторы: Li H., Gras G. Аннотация: Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, ideles, ray class fields, symbols, reciprocity laws, Hasse's principles, the Grunwald-Wang theorem, Hilbert's towers, ...). This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2003Количество страниц: 512Добавлена в каталог: 01.06.2008Операции: Положить на полку |
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152 221 236 240 243 250 293 300 441 199 217 359 210 258-260 262 271 275 305 313 237 258 266 268 304 316 342 255 228 238 244 245 301 341 371- 374 40 42 44 152 246 386 396 452 282 452 458 373 375 403 216 375 403 136 194 208 80 81 44 152 160 441 253 267 318 354 237 258 316 342 345 234 240 280 342 349 351 90 237 258 266 268 269 304 313 316 342 Abelian closure, global 163 274 279 291 323 Abelian closure, local 86 90 115 298 Absolute value 9 14 Approximation theorem 35 94 211 286 456 457 Artin, conductor 133 144 145 Artin, group 131 145 163 256 349 Artin, map 131 138 146 162 164 242 343 361 364 367 419 455 Artin, reciprocity law 113 132 Artin, symbol 132 137 138 205 350 Central classes 398 400 Chevalley's formula 180 Chinese remainder theorem 35 94 286 456 457 Class (ideal) 141 Class field theory, cohomological 108 445 Class field theory, global infinite 121 257 340 Class field theory, Jaulent's global 340 Class field theory, local infinite 86 Class field, general 143 221 235 236 398 405 433 Class field, T-ramified, S'-split 152 Class field, tower 170 172 175 176 441 445 460 462 Class group, id 28 45 113 340 Class group, ideal 37 43 45 47 164 178 221 223 283 398 402 Class group, logarithmic 215 319 355 357 Class group, ordinary sense 8 38 223 Class group, restricted sense 8 38 223 Cohomology 96 97 108 182 287 321 361-363 442 447 458 465 Completion of a number field 13 15 21 Completion of an extension 20 Completion, profinite 87 229 Completion, semi-local 331 Complexification, extension 112 139 143 146 161 167 223 241 275 280 296 419 Complexification, real place 7 24 30 140 161 365 445 Conductor, computation 81 126 127 Conductor, congruence group 164 Conductor, global 118 126 133 144 231 389 Conductor, local 80 83 Conductor, nonabelian 128 Conductor, theorem 127 Congruence group 164 Conjecture, 258-260 262 271 275 Conjecture, Gross's 215 318 358 Conjecture, Leopoldt 237 253 258 262 271 274 Conjecture, strong 305 313 Connected component 30 32 34 50 122 234 248 282 297 325 445 Cyclotomic field, global 7 137 161 168 232 233 Cyclotomic field, local 90 115 389 Decomposition (group, field) 17 19 39 66 69 71 119 140 151 224 226 287 289 299 302 304 317 333 337 341 344 Decomposition law, 237 258 316 342 345 Decomposition law, general 112 140 147 163 Decomposition law, Kummer 59 Decomposition law, wild place 300 345 Decomposition of a place in an extension 10 Density theorem 141 Deployment theorem 232 275 277 278 289 337 338 341 385 393 446 448 Dirichlet - Herbrand theorem 24 157 261 265 307 308 Dirichlet theorem 268 282 Divisible (element, group) 31 32 49 124 245 297 325 331 446 Duality 53 56 57 97 182 287 331 332 334 411 414 443 444 449 Embedding, cyclic 328 Embedding, for a 238 301 Embedding, idelic 28 41 48 235 264 Embedding, Kummer extension 351 Embedding, of a number field 12 13 15 18 66 91 Exact sequences (of global class field theory) 4 5 236 239 250 279 280 294 Exceptional case 183 190 191 411 415 427 430 431 437 439 Existence theorem, free pro- 73 234 241 257 280 284 285 300 374 452 464 Existence theorem, global 118 141 160 Frobenius automorphism, global 39 70 111 131 148 204 250 343 372 404 Frobenius automorphism, local 68 69 75 107 Fundamental class 109 121 Genera (number of) 381 383 385 389 Generalized class group 37 40 130 157 164 407 438 Genus, exact sequence 390 394 Genus, field 376-378 435 Genus, theory 375 390 408 419 Governing field 410 438 Gross's conjecture 215 318 358 Grunwald - Wang theorem 332 335 439 Hasse - Minkowski theorem 192 202 Hasse principle, defect 181 398 400 402 405 Hasse principle, general 176 Hasse principle, normic 179 180 329 402 Hasse principle, powers 183 191 Hasse principle, quadratic form 192 202 Hasse symbol, computation 135 137 Hasse symbol, general 105 106 113 116 135 207 343 395 Hensel's lemma 22 Higher ramification group 72 129 140 Hilbert symbol, global 200 201 210 358 Hilbert symbol, global computation 207 Hilbert symbol, local 85 195 200 217 Hilbert symbol, quadratic 192 Hilbert symbol, regular 198 200 Hilbert, class field 62 134 143 146 148 152 169 221 223 235 376 405 433 Hilbert, kernel 213 318 360 Hilbert, tower 170 172 175 176 441 460 Id 184 324 Id 28 45 113 340 Id 91 94 Id 28 Id 340 Id 33 48 331 Id 31 33 49 Id 30 340 Inertia (group, field) 66 69 71 119 140 150 224 287 296 302 344 Infinitesimal number 247 286 355 454 456 458 Invariant classes 178 180 365 369 Knot group 182 398 400 401 405 454 465 Kummer duality 57 156 411 Kummer theory 54 58 59 81 85 160 184 348 384 423 Leopoldt Conjecture 237 253 258 262 271 274 local 77 Local norm group (definition) 98 Logarithm 242 284 343 355 370 Martinet's constants 171 176 Modulus 23 Monogeneous 27 259 262 268 275 308 322 364 394 456 Moore's theorem 210 214 359 Norm group, Galois action 120 Norm group, global 118 120 127 142 146 160 329 398 402 405 Norm group, local 77 78 80 83 85 88 98 100 106 115 137 254 381 454 Norm group, local-global relation 111 Norm invariants (local) 100 Norm lifting theorem, global 119 142 Norm lifting theorem, local 79 Norm of class group 164 Norm of local elements 104 Norm of local units 104 Norm principle defect 181 398 400 402 405 Norm residue symbol 74-76 107 115 195 196 343 382 391 397 Norm, local-global relation 92 Ordinary class group 38 223 Ordinary sense 8 223 Ostrowski theorem 9 Place (of a number field) 7 9 11 14 Power residue symbol 204 350 Principal id 28 Principal ideal theorem 168 Product formula 112 116 135 179 201 202 206 209 219 336 356 358 387 393 395 397 Quadratic field 83 144 223 329 386 Quadratic form 180 192 202 387 Quadratic reciprocity law 113 217 Radical, initial 330 349 351 Radical, Kummer 54 58 153 156 215 411 415 418 423 433 Ramification (global), 371 373 374 Ramification (global), minimal 421 Ramification (global), tame 73 77 81 89 91 118 140 150 152 168 173 221 230 289 348 418 430 Ramification (global), wild 59 82 230 240 300 345 Ray class field, general 143 146 147 161 168 276 299 377 385 407 Ray class field, ramification index 150 Ray class field, residue degree 140 150 Ray class group 37 Ray group 23 Reciprocity law, 204 206 Reciprocity law, Artin 113 132 Reciprocity law, quadratic 113 217 Reciprocity map, global 105 108 110 122 146 148 222 244 295 324 340 411 446 Reciprocity map, local 74 88 109 115 Reflection theorem 44 152 157 214 216 283 323 374 408 Regular kernel 213 216 Residue field 21 Restricted class group 38 223 Restricted sense 8 223 S-class group 37 40 130 157 164 407 438 S-unit group, global 23 24 235 261 271 397 414 S-unit group, idelic 30 Schmidt - Chevalley theorem 124 288 340 Shapiro's Lemma 96 362 445 447 457 Signature 12 Special case 183 184 190 324 332 335 354 Spiegelungssatz 156 158 159 Splitting of a class field 404 Symbol, Artin 132 137 138 205 350 Symbol, Hasse 105 106 113 116 135 207 343 395 Symbol, Hilbert 85 195 200 201 210 217 358 Symbol, norm residue 74-76 107 115 195 196 343 358 382 391 397 Symbol, on a field 199 Symbol, power residue 204 350 Takagi group 164 Tame kernel (= regular kernel) 213 Tame ramification 73 77 81 89 91 118 140 150 152 168 173 221 230 289 348 418 430 Topological group 30 50 51 53 Topology (id 31 33 49 Torsion group ( 241 250 253 280 282 319 345 362 369 451 458 Transfer map 74 75 106 139 168 327 361 364 367 404 448 450 Unit group, global 23 180 223 261 Unit group, idelic 30 340 Unit group, local 21 230 Unit group, restricted sense 8 24 Unramified extension, global 62 139 143 148 176 302 Unramified extension, local 68 75 77 Valuation 12 355 Wild kernel (= Hilbert kernel) 213 318 360 Wild ramification 59 82 230 240 300 345
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