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Albert–Brauer–Hasse–Noether theorem

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25: 280: 54: 471:
Fenster, D.D.; Schwermer, J. (2005), "Delicate collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras",
674: 649: 628: 213: 368:, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is 545: 512: 76: 47: 620: 500: 141: 669: 574:
Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften
365: 37: 586: 41: 33: 537: 133: 90: 581: 181: 173: 129: 102: 98: 58: 410: 157: 595: 638: 603: 555: 522: 440: 417:(1932), "A determination of all normal division algebras over an algebraic number field", 8: 188: 493: 397: 392: 566: 431: 645: 624: 541: 508: 137: 634: 599: 551: 518: 480: 436: 426: 373: 591: 504: 562: 447: 145: 484: 663: 529: 455: 451: 414: 311: 153: 149: 121: 109: 536:, London Mathematical Society Monographs. New Series, vol. 28, 617:
Contributions to the history of number theory in the 20th century
458:(1932), "Beweis eines Hauptsatzes in der Theorie der Algebren", 567:"The Brauer–Hasse–Noether theorem in historical perspective" 136:
and leads to a complete description of finite-dimensional
644:
Albert, Nancy E. (2005), "A & His Algebra, iUniverse,
372:, i.e. can be obtained by an explicit construction from a 275:{\displaystyle A\otimes _{K}K_{v}\simeq M_{d}(K_{v}).} 216: 492: 446: 274: 661: 470: 46:but its sources remain unclear because it lacks 140:over algebraic number fields in terms of their 314:, one shows that two central simple algebras 619:, Heritage of European Mathematics, Zürich: 409: 163: 198:splits over the corresponding local field 585: 430: 77:Learn how and when to remove this message 614: 561: 662: 528: 490: 473:Archive for History of Exact Sciences 289:is isomorphic to the matrix algebra 18: 675:Theorems in algebraic number theory 348:are isomorphic over the completion 95:Albert–Brauer–Hasse–Noether theorem 13: 14: 686: 503:, vol. 88, New York-Berlin: 432:10.1090/s0002-9947-1932-1501659-x 330:if and only if their completions 144:. It was proved independently by 128:. The theorem is an example of a 23: 322:over an algebraic number field 305: 266: 253: 1: 621:European Mathematical Society 501:Graduate Texts in Mathematics 403: 7: 386: 10: 691: 485:10.1007/s00407-004-0093-6 655: 615:Roquette, Peter (2013), 491:Pierce, Richard (1982), 164:Statement of the theorem 108:which splits over every 32:This article includes a 16:Theorem in number theory 538:Oxford University Press 419:Trans. Amer. Math. Soc. 187:. Suppose that for any 134:algebraic number theory 91:algebraic number theory 61:more precise citations. 374:cyclic field extension 276: 182:algebraic number field 174:central simple algebra 130:local-global principle 103:algebraic number field 99:central simple algebra 460:J. reine angew. Math. 366:Grunwald–Wang theorem 277: 158:Abraham Adrian Albert 495:Associative algebras 326:are isomorphic over 310:Using the theory of 214: 670:Class field theory 613:Revised version — 398:Hasse norm theorem 393:Class field theory 364:Together with the 272: 34:list of references 650:978-0-595-32817-8 630:978-3-03719-113-2 623:, pp. 1–76, 138:division algebras 87: 86: 79: 682: 641: 612: 611: 610: 589: 571: 558: 525: 498: 487: 467: 443: 434: 281: 279: 278: 273: 265: 264: 252: 251: 239: 238: 229: 228: 142:local invariants 82: 75: 71: 68: 62: 57:this article by 48:inline citations 27: 26: 19: 690: 689: 685: 684: 683: 681: 680: 679: 660: 659: 658: 631: 608: 606: 569: 563:Roquette, Peter 548: 540:, p. 276, 515: 505:Springer-Verlag 406: 389: 356: 347: 338: 308: 297: 260: 256: 247: 243: 234: 230: 224: 220: 215: 212: 211: 206: 166: 119: 83: 72: 66: 63: 52: 38:related reading 28: 24: 17: 12: 11: 5: 688: 678: 677: 672: 657: 654: 653: 652: 642: 629: 587:10.1.1.72.4101 559: 546: 534:Maximal Orders 526: 513: 488: 479:(4): 349–379, 468: 444: 425:(3): 722–726, 405: 402: 401: 400: 395: 388: 385: 352: 343: 334: 307: 304: 293: 283: 282: 271: 268: 263: 259: 255: 250: 246: 242: 237: 233: 227: 223: 219: 202: 165: 162: 146:Richard Brauer 122:matrix algebra 115: 97:states that a 85: 84: 42:external links 31: 29: 22: 15: 9: 6: 4: 3: 2: 687: 676: 673: 671: 668: 667: 665: 651: 647: 643: 640: 636: 632: 626: 622: 618: 605: 601: 597: 593: 588: 583: 579: 575: 568: 564: 560: 557: 553: 549: 547:0-19-852673-3 543: 539: 535: 531: 527: 524: 520: 516: 514:0-387-90693-2 510: 506: 502: 497: 496: 489: 486: 482: 478: 474: 469: 465: 461: 457: 453: 449: 445: 442: 438: 433: 428: 424: 420: 416: 412: 408: 407: 399: 396: 394: 391: 390: 384: 382: 378: 375: 371: 367: 362: 360: 355: 351: 346: 342: 337: 333: 329: 325: 321: 317: 313: 303: 301: 296: 292: 288: 269: 261: 257: 248: 244: 240: 235: 231: 225: 221: 217: 210: 209: 208: 205: 201: 197: 193: 190: 186: 183: 179: 175: 171: 161: 159: 155: 151: 147: 143: 139: 135: 131: 127: 123: 118: 114: 111: 107: 104: 100: 96: 92: 81: 78: 70: 60: 56: 50: 49: 43: 39: 35: 30: 21: 20: 616: 607:, retrieved 577: 573: 533: 494: 476: 472: 463: 459: 422: 418: 411:Albert, A.A. 380: 376: 369: 363: 358: 353: 349: 344: 340: 335: 331: 327: 323: 319: 315: 312:Brauer group 309: 306:Applications 299: 294: 290: 286: 284: 203: 199: 195: 191: 184: 177: 169: 167: 154:Emmy Noether 150:Helmut Hasse 125: 116: 112: 105: 94: 88: 73: 64: 53:Please help 45: 456:Noether, E. 59:introducing 664:Categories 639:1276.11001 609:2009-07-05 604:1060.01009 556:1024.16008 530:Reiner, I. 523:0497.16001 448:Brauer, R. 441:0005.05003 404:References 357:for every 110:completion 67:April 2016 582:CiteSeerX 466:: 399–404 452:Hasse, H. 415:Hasse, H. 241:≃ 222:⊗ 189:valuation 565:(2005), 532:(2003), 387:See also 180:over an 176:of rank 101:over an 596:2222818 156:and by 55:improve 648:  637:  627:  602:  594:  584:  554:  544:  521:  511:  439:  370:cyclic 152:, and 93:, the 656:Notes 570:(PDF) 285:Then 172:be a 124:over 120:is a 40:, or 646:ISBN 625:ISBN 542:ISBN 509:ISBN 339:and 318:and 168:Let 635:Zbl 600:Zbl 552:Zbl 519:Zbl 481:doi 464:167 437:Zbl 427:doi 302:). 132:in 89:In 666:: 633:, 598:, 592:MR 590:, 580:, 578:15 576:, 572:, 550:, 517:, 507:, 499:, 477:59 475:, 462:, 454:; 450:; 435:, 423:34 421:, 413:; 383:. 361:. 207:: 194:, 160:. 148:, 44:, 36:, 483:: 429:: 381:K 379:/ 377:L 359:v 354:v 350:K 345:v 341:B 336:v 332:A 328:K 324:K 320:B 316:A 300:K 298:( 295:d 291:M 287:A 270:. 267:) 262:v 258:K 254:( 249:d 245:M 236:v 232:K 226:K 218:A 204:v 200:K 196:A 192:v 185:K 178:d 170:A 126:K 117:v 113:K 106:K 80:) 74:( 69:) 65:( 51:.

Index

list of references
related reading
external links
inline citations
improve
introducing
Learn how and when to remove this message
algebraic number theory
central simple algebra
algebraic number field
completion
matrix algebra
local-global principle
algebraic number theory
division algebras
local invariants
Richard Brauer
Helmut Hasse
Emmy Noether
Abraham Adrian Albert
central simple algebra
algebraic number field
valuation
Brauer group
Grunwald–Wang theorem
cyclic field extension
Class field theory
Hasse norm theorem
Albert, A.A.
Hasse, H.

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