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Brauer–Siegel theorem

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514: 400: 270: 108: 298: 462: 579: 159: 555: 177: 472: 475:, as indeed was the result on quadratic fields on which it built. Effective results in the same direction were initiated in work of 574: 497: 584: 548: 48: 118: 146: 52: 541: 61: 36: 468:, then one may drop the assumption of normality - this is what is actually proved in Brauer's paper. 44: 32: 395:{\displaystyle {\frac {\log(h_{i}R_{i})}{\log {\sqrt {|D_{i}|}}}}\to 1{\text{ as }}i\to \infty } 285: 525: 488: 40: 28: 426: 568: 521: 513: 476: 20: 149:. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if 265:{\displaystyle {\frac {}{\log |D_{i}|}}\to 0{\text{ as }}i\to \infty .} 16:
Asymptotic result on the behaviour of algebraic number fields
529: 429: 301: 180: 64: 47:. It attempts to generalise the results known on the 456: 394: 264: 102: 275:Assuming that, and the algebraic hypothesis that 493:On the Zeta-Function of Algebraic Number Fields 55:, to a more general sequence of number fields 35:, is an asymptotic result on the behaviour of 549: 8: 113:In all cases other than the rational field 556: 542: 446: 437: 428: 378: 362: 356: 347: 345: 328: 318: 302: 300: 245: 231: 225: 216: 200: 191: 181: 179: 82: 69: 63: 464:are bounded above by a uniform constant 423:. If one assumes that all the degrees 103:{\displaystyle K_{1},K_{2},\ldots .\ } 7: 510: 508: 145:then has units of infinite order by 138:must be taken into account, because 117:and imaginary quadratic fields, the 580:Theorems in algebraic number theory 389: 256: 14: 512: 447: 201: 498:American Journal of Mathematics 451: 430: 386: 372: 363: 348: 334: 311: 253: 239: 232: 217: 205: 184: 1: 528:. You can help Knowledge by 601: 507: 53:imaginary quadratic fields 292:, the conclusion is that 147:Dirichlet's unit theorem 414:is the class number of 37:algebraic number fields 575:Analytic number theory 524:-related article is a 479:from the early 1970s. 458: 396: 266: 104: 459: 397: 267: 105: 25:Brauer–Siegel theorem 427: 299: 178: 62: 585:Number theory stubs 501:69 (1947), 243–250. 454: 392: 262: 100: 45:Carl Ludwig Siegel 33:Carl Ludwig Siegel 537: 536: 381: 370: 367: 248: 237: 99: 592: 558: 551: 544: 516: 509: 463: 461: 460: 457:{\displaystyle } 455: 450: 442: 441: 401: 399: 398: 393: 382: 379: 371: 369: 368: 366: 361: 360: 351: 346: 337: 333: 332: 323: 322: 303: 286:Galois extension 271: 269: 268: 263: 249: 246: 238: 236: 235: 230: 229: 220: 208: 204: 196: 195: 182: 109: 107: 106: 101: 97: 87: 86: 74: 73: 600: 599: 595: 594: 593: 591: 590: 589: 565: 564: 563: 562: 505: 485: 471:This result is 433: 425: 424: 422: 413: 352: 338: 324: 314: 304: 297: 296: 283: 221: 209: 187: 183: 176: 175: 170: 157: 144: 137: 128: 78: 65: 60: 59: 17: 12: 11: 5: 598: 596: 588: 587: 582: 577: 567: 566: 561: 560: 553: 546: 538: 535: 534: 517: 503: 502: 489:Richard Brauer 484: 481: 453: 449: 445: 440: 436: 432: 418: 409: 403: 402: 391: 388: 385: 380: as  377: 374: 365: 359: 355: 350: 344: 341: 336: 331: 327: 321: 317: 313: 310: 307: 279: 273: 272: 261: 258: 255: 252: 247: as  244: 241: 234: 228: 224: 219: 215: 212: 207: 203: 199: 194: 190: 186: 166: 153: 142: 133: 124: 111: 110: 96: 93: 90: 85: 81: 77: 72: 68: 41:Richard Brauer 39:, obtained by 29:Richard Brauer 27:, named after 15: 13: 10: 9: 6: 4: 3: 2: 597: 586: 583: 581: 578: 576: 573: 572: 570: 559: 554: 552: 547: 545: 540: 539: 533: 531: 527: 523: 522:number theory 518: 515: 511: 506: 500: 499: 494: 490: 487: 486: 482: 480: 478: 474: 469: 467: 443: 438: 434: 421: 417: 412: 408: 383: 375: 357: 353: 342: 339: 329: 325: 319: 315: 308: 305: 295: 294: 293: 291: 287: 282: 278: 259: 250: 242: 226: 222: 213: 210: 197: 192: 188: 174: 173: 172: 169: 165: 161: 156: 152: 148: 141: 136: 132: 127: 123: 120: 116: 94: 91: 88: 83: 79: 75: 70: 66: 58: 57: 56: 54: 50: 49:class numbers 46: 42: 38: 34: 30: 26: 22: 530:expanding it 519: 504: 496: 492: 477:Harold Stark 470: 465: 419: 415: 410: 406: 404: 289: 280: 276: 274: 167: 163: 160:discriminant 154: 150: 139: 134: 130: 125: 121: 114: 112: 24: 18: 473:ineffective 21:mathematics 569:Categories 483:References 390:∞ 387:→ 373:→ 343:⁡ 309:⁡ 257:∞ 254:→ 240:→ 214:⁡ 119:regulator 92:… 171:, then 158:is the 405:where 98:  23:, the 520:This 284:is a 526:stub 43:and 31:and 340:log 306:log 288:of 211:log 162:of 129:of 51:of 19:In 571:: 495:, 491:, 557:e 550:t 543:v 532:. 466:N 452:] 448:Q 444:: 439:i 435:K 431:[ 420:i 416:K 411:i 407:h 384:i 376:1 364:| 358:i 354:D 349:| 335:) 330:i 326:R 320:i 316:h 312:( 290:Q 281:i 277:K 260:. 251:i 243:0 233:| 227:i 223:D 218:| 206:] 202:Q 198:: 193:i 189:K 185:[ 168:i 164:K 155:i 151:D 143:i 140:K 135:i 131:K 126:i 122:R 115:Q 95:. 89:, 84:2 80:K 76:, 71:1 67:K

Index

mathematics
Richard Brauer
Carl Ludwig Siegel
algebraic number fields
Richard Brauer
Carl Ludwig Siegel
class numbers
imaginary quadratic fields
regulator
Dirichlet's unit theorem
discriminant
Galois extension
ineffective
Harold Stark
Richard Brauer
American Journal of Mathematics
Stub icon
number theory
stub
expanding it
v
t
e
Categories
Analytic number theory
Theorems in algebraic number theory
Number theory stubs

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