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Octic reciprocity

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520: 397: 196: 248: 97: 580: 122: 392:{\displaystyle \left({\frac {p}{q}}\right)_{8}\left({\frac {q}{p}}\right)_{8}=\left({\frac {aB-bA}{q}}\right)_{4}\left({\frac {cD-dC}{q}}\right)_{2}\ .} 561: 437: 468: 585: 554: 39: 60: 54: 547: 413: 47: 32: 487: 433: 408: 43: 503: 477: 451: 499: 447: 507: 495: 455: 443: 28: 531: 574: 527: 20: 519: 191:{\displaystyle \left({\frac {p}{q}}\right)_{4}=\left({\frac {q}{p}}\right)_{4}=+1.} 35: 463: 432:, Springer Monographs in Mathematics, Springer-Verlag, Berlin, pp. 289–316, 427: 491: 482: 16:
Reciprocity law relating the residues of 8th powers modulo primes
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be distinct primes congruent to 1 modulo 8, such that
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for 8th powers, due to Williams. Define the symbol
391: 190: 91: 92:{\displaystyle \left({\frac {x}{p}}\right)_{k}} 555: 8: 562: 548: 429:Reciprocity laws. From Euler to Eisenstein 481: 377: 349: 338: 310: 296: 282: 271: 257: 250: 173: 159: 145: 131: 124: 83: 69: 62: 7: 516: 514: 31:relating the residues of 8th powers 581:Theorems in algebraic number theory 464:"A rational octic reciprocity law" 14: 518: 469:Pacific Journal of Mathematics 1: 462:Williams, Kenneth S. (1976), 534:. You can help Knowledge by 40:law of quadratic reciprocity 426:Lemmermeyer, Franz (2000), 107:-th power modulo the prime 602: 513: 55:rational reciprocity law 483:10.2140/pjm.1976.63.563 111:and -1 otherwise. Let 530:-related article is a 414:Eisenstein reciprocity 393: 192: 93: 394: 193: 94: 249: 123: 61: 586:Number theory stubs 48:quartic reciprocity 38:, analogous to the 389: 188: 89: 543: 542: 409:Artin reciprocity 385: 371: 332: 290: 265: 167: 139: 77: 44:cubic reciprocity 25:octic reciprocity 593: 564: 557: 550: 522: 515: 510: 485: 458: 398: 396: 395: 390: 383: 382: 381: 376: 372: 367: 350: 343: 342: 337: 333: 328: 311: 301: 300: 295: 291: 283: 276: 275: 270: 266: 258: 197: 195: 194: 189: 178: 177: 172: 168: 160: 150: 149: 144: 140: 132: 98: 96: 95: 90: 88: 87: 82: 78: 70: 601: 600: 596: 595: 594: 592: 591: 590: 571: 570: 569: 568: 461: 440: 425: 422: 405: 351: 345: 344: 312: 306: 305: 278: 277: 253: 252: 247: 246: 155: 154: 127: 126: 121: 120: 65: 64: 59: 58: 29:reciprocity law 17: 12: 11: 5: 599: 597: 589: 588: 583: 573: 572: 567: 566: 559: 552: 544: 541: 540: 523: 512: 511: 476:(2): 563–570, 459: 438: 421: 418: 417: 416: 411: 404: 401: 400: 399: 388: 380: 375: 370: 366: 363: 360: 357: 354: 348: 341: 336: 331: 327: 324: 321: 318: 315: 309: 304: 299: 294: 289: 286: 281: 274: 269: 264: 261: 256: 187: 184: 181: 176: 171: 166: 163: 158: 153: 148: 143: 138: 135: 130: 86: 81: 76: 73: 68: 15: 13: 10: 9: 6: 4: 3: 2: 598: 587: 584: 582: 579: 578: 576: 565: 560: 558: 553: 551: 546: 545: 539: 537: 533: 529: 528:number theory 524: 521: 517: 509: 505: 501: 497: 493: 489: 484: 479: 475: 471: 470: 465: 460: 457: 453: 449: 445: 441: 439:3-540-66957-4 435: 431: 430: 424: 423: 419: 415: 412: 410: 407: 406: 402: 386: 378: 373: 368: 364: 361: 358: 355: 352: 346: 339: 334: 329: 325: 322: 319: 316: 313: 307: 302: 297: 292: 287: 284: 279: 272: 267: 262: 259: 254: 245: 244: 243: 241: 237: 233: 229: 225: 221: 217: 213: 209: 205: 201: 185: 182: 179: 174: 169: 164: 161: 156: 151: 146: 141: 136: 133: 128: 118: 114: 110: 106: 102: 84: 79: 74: 71: 66: 56: 51: 49: 45: 41: 37: 34: 30: 26: 22: 21:number theory 536:expanding it 525: 473: 467: 428: 239: 235: 231: 227: 223: 219: 215: 211: 207: 203: 199: 116: 112: 108: 104: 100: 99:to be +1 if 52: 24: 18: 242:odd. Then 53:There is a 575:Categories 508:0311.10004 456:0949.11002 420:References 492:0030-8730 359:− 320:− 403:See also 500:0414467 448:1761696 238:, with 506:  498:  490:  454:  446:  436:  384:  46:, and 36:primes 33:modulo 526:This 103:is a 27:is a 532:stub 488:ISSN 434:ISBN 218:and 198:Let 115:and 504:Zbl 478:doi 452:Zbl 234:+ 2 214:+ 2 19:In 577:: 502:, 496:MR 494:, 486:, 474:63 472:, 466:, 450:, 444:MR 442:, 240:aA 230:= 226:+ 222:= 210:= 206:+ 202:= 186:1. 50:. 42:, 23:, 563:e 556:t 549:v 538:. 480:: 387:. 379:2 374:) 369:q 365:C 362:d 356:D 353:c 347:( 340:4 335:) 330:q 326:A 323:b 317:B 314:a 308:( 303:= 298:8 293:) 288:p 285:q 280:( 273:8 268:) 263:q 260:p 255:( 236:D 232:C 228:B 224:A 220:q 216:d 212:c 208:b 204:a 200:p 183:+ 180:= 175:4 170:) 165:p 162:q 157:( 152:= 147:4 142:) 137:q 134:p 129:( 117:q 113:p 109:p 105:k 101:x 85:k 80:) 75:p 72:x 67:(

Index

number theory
reciprocity law
modulo
primes
law of quadratic reciprocity
cubic reciprocity
quartic reciprocity
rational reciprocity law
Artin reciprocity
Eisenstein reciprocity
Reciprocity laws. From Euler to Eisenstein
ISBN
3-540-66957-4
MR
1761696
Zbl
0949.11002
"A rational octic reciprocity law"
Pacific Journal of Mathematics
doi
10.2140/pjm.1976.63.563
ISSN
0030-8730
MR
0414467
Zbl
0311.10004
Stub icon
number theory
stub

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