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Abel's irreducibility theorem

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452: 378:. Irreducible polynomials, divided by their leading coefficient, are minimal for their roots (Cox Proposition 4.1.5), and all minimal polynomials are irreducible, so Cox's formulation is equivalent to Abel's. 238: 214: 493: 397: 486: 371: 355: 517: 479: 512: 417: 28: 177: 67: 55: 306: 48: 323: 219: 195: 428: 425: 393: 351: 345: 297: 32: 463: 459: 385: 315: 276: 189: 181: 240:
as a root. Furthermore, there is no same-degree polynomial that shares any roots with
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as a root; hence there is no linear or constant polynomial over the rationals having
451: 161: 151: 20: 44: 433: 319: 302:"Mémoire sur une classe particulière d'équations résolubles algébriquement" 389: 304:[Note on a particular class of algebraically solvable equations], 347:
100 Great Problems of Elementary Mathematics: Their History and Solution
384:, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, 374:
rather than irreducible polynomials more generally, is Lemma 4.1.3 of
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Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility
467: 222: 198: 232: 208: 307:Journal für die reine und angewandte Mathematik 487: 188: − 2 is irreducible over the 184:) that shares any root with it. For example, 8: 350:, Courier Dover Publications, p. 120, 494: 480: 223: 221: 199: 197: 339: 337: 289: 7: 448: 446: 248:), other than constant multiples of 176:) is irreducible, there is no lower- 375: 466:. You can help Knowledge (XXG) by 14: 450: 275:) are two different irreducible 98:) shares at least one root with 429:"Abel's Irreducibility Theorem" 1: 25:Abel's irreducibility theorem 31:result described in 1829 by 279:, then they share no roots. 233:{\displaystyle {\sqrt {2}}} 209:{\displaystyle {\sqrt {2}}} 180:polynomial (other than the 534: 445: 344:Dörrie, Heinrich (1965), 164:of the theorem include: 320:10.1515/crll.1829.4.131 110:is divisible evenly by 518:Abstract algebra stubs 462:-related article is a 380:Cox, David A. (2012), 234: 210: 390:10.1002/9781118218457 235: 211: 126:) can be factored as 74:, then every root of 420:. September 4, 2008. 267:) ≠  220: 196: 90:). Equivalently, if 513:Field (mathematics) 372:minimal polynomials 16:Field theory result 426:Weisstein, Eric W. 370:This theorem, for 230: 206: 58:with a polynomial 35:, asserts that if 475: 474: 399:978-1-118-07205-9 277:monic polynomials 228: 204: 33:Niels Henrik Abel 525: 496: 489: 482: 460:abstract algebra 454: 447: 439: 438: 404: 402: 368: 362: 360: 341: 332: 330: 294: 239: 237: 236: 231: 229: 224: 215: 213: 212: 207: 205: 200: 190:rational numbers 118:), meaning that 533: 532: 528: 527: 526: 524: 523: 522: 503: 502: 501: 500: 443: 424: 423: 416:Larry Freeman. 413: 408: 407: 400: 379: 369: 365: 358: 343: 342: 335: 296: 295: 291: 286: 218: 217: 194: 193: 182:zero polynomial 82:) is a root of 17: 12: 11: 5: 531: 529: 521: 520: 515: 505: 504: 499: 498: 491: 484: 476: 473: 472: 455: 441: 440: 421: 412: 411:External links 409: 406: 405: 398: 363: 356: 333: 314:(4): 131–156, 288: 287: 285: 282: 281: 280: 257: 227: 203: 150:) also having 54:that shares a 15: 13: 10: 9: 6: 4: 3: 2: 530: 519: 516: 514: 511: 510: 508: 497: 492: 490: 485: 483: 478: 477: 471: 469: 465: 461: 456: 453: 449: 444: 436: 435: 430: 427: 422: 419: 415: 414: 410: 401: 395: 391: 387: 383: 382:Galois Theory 377: 373: 367: 364: 359: 357:9780486613482 353: 349: 348: 340: 338: 334: 329: 325: 321: 317: 313: 309: 308: 303: 299: 293: 290: 283: 278: 274: 270: 266: 262: 258: 255: 251: 247: 243: 225: 201: 191: 187: 183: 179: 175: 171: 167: 166: 165: 163: 159: 157: 153: 149: 145: 141: 137: 133: 129: 125: 121: 117: 113: 109: 105: 101: 97: 93: 89: 85: 81: 77: 73: 69: 65: 61: 57: 53: 50: 46: 42: 38: 34: 30: 26: 22: 468:expanding it 457: 442: 432: 381: 366: 346: 311: 305: 292: 272: 268: 264: 260: 253: 249: 245: 241: 185: 173: 169: 160: 155: 152:coefficients 147: 143: 139: 135: 131: 127: 123: 119: 115: 111: 107: 103: 99: 95: 91: 87: 83: 79: 75: 71: 63: 59: 51: 40: 36: 29:field theory 24: 18: 298:Abel, N. H. 162:Corollaries 68:irreducible 66:) that is 21:mathematics 507:Categories 376:Cox (2012) 284:References 70:over  45:polynomial 434:MathWorld 328:121388045 300:(1829), 192:and has 154:in  142:) with 106:) then 47:over a 43:) is a 396:  354:  326:  261:ƒ 250:ƒ 242:ƒ 178:degree 170:ƒ 120:ƒ 108:ƒ 92:ƒ 84:ƒ 37:ƒ 458:This 324:S2CID 49:field 464:stub 394:ISBN 352:ISBN 312:1829 56:root 27:, a 386:doi 316:doi 259:If 168:If 19:In 509:: 431:. 392:, 336:^ 322:, 310:, 256:). 158:. 23:, 495:e 488:t 481:v 470:. 437:. 403:. 388:: 361:. 331:. 318:: 273:x 271:( 269:g 265:x 263:( 254:x 252:( 246:x 244:( 226:2 202:2 186:x 174:x 172:( 156:F 148:x 146:( 144:h 140:x 138:( 136:h 134:) 132:x 130:( 128:g 124:x 122:( 116:x 114:( 112:g 104:x 102:( 100:g 96:x 94:( 88:x 86:( 80:x 78:( 76:g 72:F 64:x 62:( 60:g 52:F 41:x 39:(

Index

mathematics
field theory
Niels Henrik Abel
polynomial
field
root
irreducible
coefficients
Corollaries
degree
zero polynomial
rational numbers
monic polynomials
Abel, N. H.
"Mémoire sur une classe particulière d'équations résolubles algébriquement"
Journal für die reine und angewandte Mathematik
doi
10.1515/crll.1829.4.131
S2CID
121388045


100 Great Problems of Elementary Mathematics: Their History and Solution
ISBN
9780486613482
minimal polynomials
Cox (2012)
doi
10.1002/9781118218457
ISBN

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