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Li H., Gras G. — Class Field Theory: From Theory to Practice
Li H., Gras G. — Class Field Theory: From Theory to Practice

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Название: Class Field Theory: From Theory to Practice

Авторы: 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: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 512

Добавлена в каталог: 01.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$H^{S}_{T}$, $H^{ord}_{p}$, $H^{res}_{ta}$      152 221 236 240 243 250 293 300 441
$K_{2}$ of a field      199 217 359
$K_{2}$, fundamental diagram)      210
$p$-adic conjecture      258-260 262 271 275 305 313
$p$-adic rank      237 258 266 268 304 316 342
$p$-adic zeta function      255
$p$-completion      228 238 244 245 301 341
$p$-primitive (set, extension)      371- 374
$p$-rank, class group      40 42 44 152 246 386 396
$p$-rank, cohomology group      452
$p$-rank, torsion group      282 452 458
$p$-rational field      373 375 403
$p$-regular field      216 375 403
$v$-associate      136 194 208
$v$-conductor      80 81
$\hat{S}$afarevi$\hat{c}$'s formula      44 152 160 441
$\mathbb{Z}_{p}$-extension, cyclotomic      253 267 318 354
$\mathbb{Z}_{p}$-extension, decomposition law      237 258 316 342 345
$\mathbb{Z}_{p}$-extension, general      234 240 280 342
$\mathbb{Z}_{p}$-extension, initial radical      349 351
$\mathbb{Z}_{p}$-extension, local      90
$\mathbb{Z}_{p}$-rank (Galois group)      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$\grave{e}$le      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, $p$-adic      258-260 262 271 275
Conjecture, Gross's $p$-adic      215 318 358
Conjecture, Leopoldt      237 253 258 262 271 274
Conjecture, strong $p$-adic      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, $\mathbb{Z}_{p}$-extension      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 $p$-completion      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-$p$-extension      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$\grave{e}$le class (of finite order)      184 324
Id$\grave{e}$le class group      28 45 113 340
Id$\grave{e}$le group, Galois action      91 94
Id$\grave{e}$le group, general      28
Id$\grave{e}$le group, globalized      340
Id$\grave{e}$le group, reduced      33 48 331
Id$\grave{e}$le group, topology      31 33 49
Id$\grave{e}$le group, units      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$\grave{e}$le      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), $p$-primitive      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, $n$th      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$\grave{e}$le group)      31 33 49
Torsion group ($\mathcal{T}^{S}_{T}$ , $\mathcal{T}^{ord}_{p}$)      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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