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Minkowski's bound

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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle
150: 101: 187: 731: 520:{\displaystyle {\sqrt {|D|}}\geq \left({\frac {\pi }{4}}\right)^{r_{2}}{\frac {n^{n}}{n!}}\geq \left({\frac {\pi }{4}}\right)^{n/2}{\frac {n^{n}}{n!}}\ .} 70: 318:
Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence, and further, the
668: 633: 599: 625: 736: 660: 31: 555: 543: 106: 693: 84: 715: 284:{\displaystyle M_{K}={\sqrt {|D|}}\left({\frac {4}{\pi }}\right)^{r_{2}}{\frac {n!}{n^{n}}}\ .} 664: 629: 595: 319: 166: 153: 54: 43: 674: 639: 605: 678: 652: 643: 609: 591: 174: 162: 338:
Minkowski's bound may be used to derive a lower bound for the discriminant of a field
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at least 2, it is easy to show that the lower bound is greater than 1, so we obtain
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Limits ideals to be checked in order to determine the class number of a number field
47: 323: 39: 17: 698: 617: 178: 542:, is non-trivial. This implies that the field of rational numbers has no 659:. Encyclopedia of Mathematics and its Applications. Vol. 30. 590:. Encycl. Math. Sci. Vol. 62 (2nd printing of 1st ed.). 360:. Since an integral ideal has norm at least one, we have 1 ≤ 42:
of the norm of ideals to be checked in order to determine the
538:, that the discriminant of every number field, other than 628:. Vol. 110 (second ed.). New York: Springer. 378: 190: 109: 87: 694:"Using Minkowski's Constant To Find A Class Number" 519: 283: 144: 95: 8: 157: 495: 489: 479: 475: 461: 437: 431: 423: 418: 404: 389: 381: 379: 377: 267: 253: 245: 240: 226: 214: 206: 204: 195: 189: 136: 117: 108: 89: 88: 86: 573: 571: 567: 7: 53:. It is named for the mathematician 732:Theorems in algebraic number theory 657:Algorithmic Algebraic Number Theory 577:Pohst & Zassenhaus (1989) p.384 25: 181:not exceeding Minkowski's bound 554:The result is a consequence of 390: 382: 215: 207: 145:{\displaystyle 2r_{2}=n-r_{1}} 1: 626:Graduate Texts in Mathematics 96:{\displaystyle \mathbb {Q} } 753: 718:at Secret Blogging Seminar 661:Cambridge University Press 165:. Then every class in the 622:Algebraic Number Theory 588:Algebraic Number Theory 32:algebraic number theory 521: 285: 146: 97: 586:Koch, Helmut (1997). 522: 286: 147: 98: 706:Stevenhagen, Peter. 544:unramified extension 376: 322:is generated by the 295:Minkowski's constant 188: 107: 85: 716:The Minkowski Bound 556:Minkowski's theorem 536:Minkowski's Theorem 517: 281: 160: 154:complex embeddings 142: 93: 737:Hermann Minkowski 513: 509: 469: 451: 412: 394: 320:ideal class group 277: 273: 234: 219: 167:ideal class group 161:is the number of 152:be the number of 77:be the degree of 55:Hermann Minkowski 36:Minkowski's bound 16:(Redirected from 744: 703: 682: 647: 613: 578: 575: 526: 524: 523: 518: 511: 510: 508: 500: 499: 490: 488: 487: 483: 474: 470: 462: 452: 450: 442: 441: 432: 430: 429: 428: 427: 417: 413: 405: 395: 393: 385: 380: 326:of norm at most 290: 288: 287: 282: 275: 274: 272: 271: 262: 254: 252: 251: 250: 249: 239: 235: 227: 220: 218: 210: 205: 200: 199: 151: 149: 148: 143: 141: 140: 122: 121: 102: 100: 99: 94: 92: 21: 752: 751: 747: 746: 745: 743: 742: 741: 722: 721: 692: 689: 671: 650: 636: 616: 602: 592:Springer-Verlag 585: 582: 581: 576: 569: 564: 552: 501: 491: 457: 456: 443: 433: 419: 400: 399: 374: 373: 368: 359: 352: 334: 316: 309: 263: 255: 241: 222: 221: 191: 186: 185: 163:real embeddings 132: 113: 105: 104: 83: 82: 63: 28: 23: 22: 18:Minkowski bound 15: 12: 11: 5: 750: 748: 740: 739: 734: 724: 723: 720: 719: 713: 704: 688: 687:External links 685: 684: 683: 669: 653:Zassenhaus, H. 648: 634: 614: 600: 580: 579: 566: 565: 563: 560: 551: 548: 528: 527: 516: 507: 504: 498: 494: 486: 482: 478: 473: 468: 465: 460: 455: 449: 446: 440: 436: 426: 422: 416: 411: 408: 403: 398: 392: 388: 384: 364: 357: 350: 330: 315: 312: 305: 301:is this bound 297:for the field 292: 291: 280: 270: 266: 261: 258: 248: 244: 238: 233: 230: 225: 217: 213: 209: 203: 198: 194: 175:integral ideal 139: 135: 131: 128: 125: 120: 116: 112: 91: 73:of the field, 62: 59: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 749: 738: 735: 733: 730: 729: 727: 717: 714: 712: 710: 705: 701: 700: 695: 691: 690: 686: 680: 676: 672: 670:0-521-33060-2 666: 662: 658: 654: 649: 645: 641: 637: 635:0-387-94225-4 631: 627: 623: 619: 615: 611: 607: 603: 601:3-540-63003-1 597: 593: 589: 584: 583: 574: 572: 568: 561: 559: 557: 549: 547: 545: 541: 537: 533: 514: 505: 502: 496: 492: 484: 480: 476: 471: 466: 463: 458: 453: 447: 444: 438: 434: 424: 420: 414: 409: 406: 401: 396: 386: 372: 371: 370: 367: 363: 356: 349: 345: 341: 336: 333: 329: 325: 321: 313: 311: 308: 304: 300: 296: 278: 268: 264: 259: 256: 246: 242: 236: 231: 228: 223: 211: 201: 196: 192: 184: 183: 182: 180: 176: 172: 168: 164: 155: 137: 133: 129: 126: 123: 118: 114: 110: 80: 76: 72: 68: 60: 58: 56: 52: 49: 45: 41: 37: 33: 19: 709:Number Rings 708: 697: 656: 621: 587: 553: 539: 535: 531: 529: 365: 361: 354: 347: 343: 339: 337: 331: 327: 324:prime ideals 317: 306: 302: 298: 294: 293: 173:contains an 170: 78: 74: 71:discriminant 66: 64: 50: 48:number field 44:class number 35: 29: 651:Pohst, M.; 618:Lang, Serge 40:upper bound 726:Categories 699:PlanetMath 679:0685.12001 644:0811.11001 610:0819.11044 562:References 369:, so that 314:Properties 61:Definition 464:π 454:≥ 407:π 397:≥ 232:π 130:− 38:gives an 655:(1989). 620:(1994). 69:be the 677:  667:  642:  632:  608:  598:  512:  342:given 276:  156:where 103:, and 550:Proof 81:over 46:of a 665:ISBN 630:ISBN 596:ISBN 530:For 353:and 179:norm 159:r_1} 65:Let 675:Zbl 640:Zbl 606:Zbl 558:. 177:of 169:of 30:In 728:: 696:. 673:. 663:. 638:. 624:. 604:. 594:. 570:^ 546:. 346:, 335:. 310:. 57:. 34:, 711:. 702:. 681:. 646:. 612:. 540:Q 532:n 515:. 506:! 503:n 497:n 493:n 485:2 481:/ 477:n 472:) 467:4 459:( 448:! 445:n 439:n 435:n 425:2 421:r 415:) 410:4 402:( 391:| 387:D 383:| 366:K 362:M 358:2 355:r 351:1 348:r 344:n 340:K 332:K 328:M 307:K 303:M 299:K 279:. 269:n 265:n 260:! 257:n 247:2 243:r 237:) 229:4 224:( 216:| 212:D 208:| 202:= 197:K 193:M 171:K 138:1 134:r 127:n 124:= 119:2 115:r 111:2 90:Q 79:K 75:n 67:D 51:K 20:)

Index

Minkowski bound
algebraic number theory
upper bound
class number
number field
Hermann Minkowski
discriminant
complex embeddings
real embeddings
ideal class group
integral ideal
norm
ideal class group
prime ideals
unramified extension
Minkowski's theorem


Springer-Verlag
ISBN
3-540-63003-1
Zbl
0819.11044
Lang, Serge
Graduate Texts in Mathematics
ISBN
0-387-94225-4
Zbl
0811.11001
Zassenhaus, H.

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