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are provided. Moreover, this editor describes (in an edit summary) his edit as "I made everything very carefully. Read the
Hermite essay in the Italian version on page 258! These is this special formula. And I only did regular calculation algorithms we all know. And the last step I wrote down in this
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Every equation may be solved using a fifth root. It suffice to replace any number appearing in the solution by the fifth power of its fifth root. This may seem a joke, but Galois theory allows showing that fifth roots (if any) may always be eliminated from any solution of an equation of lower degree.
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Unfortunately I am unable to find the relation between the root z in the Cayley's resolvent and the actual root y of the depressed quintic equation - the one lacking the biquadratic term in y! I scanned several times the related paragraphs in the main wiki-article but couldn't see a given relation.
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There are other reasons to remove this section: firstly, notations are not defined, nothing is said on the relation of the first formula with the general quintic, and the final result is not explained. This is not acceptable for an encyclopedia. Also, Hermite's work is mentioned in the preceding
1661:), this new parameterization cannot be inserted in Knowledge if it has not been published in a referred and notable academic journal. Ever if it has been reliably published, it is not necessary worth to mention it in Knowledge: this would need that it would be cited in secondary sources (see
1319:{\displaystyle k_{1}={\frac {{b}^{2}\,\left({b}^{2}\,g-{c}^{3}\right)\,\left({b}^{4}\,{g}^{2}+{c}^{6}\right)\,\left({b}^{8}\,{g}^{4}-22\,{b}^{6}\,{c}^{3}\,{g}^{3}-6\,{b}^{4}\,{c}^{6}\,{g}^{2}+22\,{b}^{2}\,{c}^{9}\,g+{c}^{12}\right)}{{c}^{2}\,{g}^{4}\,{\left({b}^{2}\,g+{c}^{3}\right)}^{4}}}}
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This means that it is difficult to find the relationship between the rational root of Cayley's resolvent and the roots of the quintic. In fact, I have given a solution to this problem (cited in the article), and the description of the formula is three pages length.
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Cayley's result allows us to test if a quintic is solvable. If it is the case, finding its roots is a more difficult problem, which consists of expressing the roots in terms of radicals involving the coefficients of the quintic and the rational root of Cayley's
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This formula seems a new parameterization of solvable quintics in Bring–Jerrard form. The article gives already three different such parameterizations, which are all simpler than yours. As
Knowledge is not a media for publishing original research (see
896:{\displaystyle k_{2}={\frac {5\,{b}^{2}\,\left({b}^{2}\,g-{c}^{3}\right)\,\left({b}^{4}\,{g}^{2}-{b}^{2}\,{c}^{3}\,g-{c}^{6}\right)\,\left({b}^{4}\,{g}^{2}+4\,{b}^{2}\,{c}^{3}\,g-{c}^{6}\right)}{c\,{g}^{3}\,{\left({b}^{2}\,g+{c}^{3}\right)}^{3}}}}
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Article reads: "the
Tschirnhaus transformation ...which depresses the quintic". I bet most readers have no idea what it means to "depress" a quintic. Some more explanation would be helpful (or maybe even an article on depressing polynomials?).
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By the way, a similar move would be useful for cubic, quartic, and other, as in all these articles the "function" aspect is minor, and most of the content is devoted to the equations, which are studied using only polynomial technics.
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455:, and rewording the sentence, but this results in a wrong assertion (at least if there is no typo in the formula for Cayley's resolvent). As no source is provided, which allows to correct this sentence, I'll remove it as .
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Should this start out as "If an irreducible quintic is solvable..."? I ask because a quintic that can be factored into a cubic and a quadratic can be solved without fifth roots—can it also be solved using a fifth root?
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The beginning of the section already said that "solvable quintics" refer to irreducible quintics solving by radicals. I have clarified this by adding that only irreducible quintics are considered in the section.
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article just needs the theorems of the lemniscate elliptic functions that are also known already". This is exactly what is called "original research" in
Knowledge. So, the Knowledge policy
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Can you help me see if there is any of such relation given in the article? If not, please add it immediately. I wait for your reply in my e-mail account. Yours truly,
2042:. The problem is that somebody has decided that "Quintic function" should be different from "Quintic equation". Some time ago, I have encountered this problem with
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If the quintic is solvable, one of the solutions may be represented by an algebraic expression involving a fifth root and at most two square roots, generally nested.
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for details. In the linked article, the relation of the subject with elliptic integrals is detailed and the other authors who have contributed are mentioned.
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Insists to add a section specifically on
Hermite's work on the subject. This section is not convenient for this Knowledge article. Firstly, it seems to be an
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PS to my preceding post: There is no evidence that above parameterized set of quintics is solvable. A proof of solvability could be an expression of
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The first external link to the
Quintic Equation Solver appears to be broken, or the solver isn't working. Page comes up blank.
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1665:), or, at least that it is evident that would improve the article. In any case, because of your conflict of interest (see
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In other words, this new section is a poor duplication of an existing content. This is a further reason for removing it.
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2068:(made your page in english visible from there), but this page in english is still no longer visible from, for instance,
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1600:{\displaystyle g\,{m}^{\frac {4}{5}}-{\frac {b\,g\,{m}^{\frac {3}{5}}}{c}}+c\,{m}^{\frac {2}{5}}+b\,{m}^{\frac {1}{5}}}
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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Before my recent edits, the article contained the following assertion, which I quote verbatim for the record:
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You fixed the problem here (made the corresponding pages in other languages visible from here), and so did I
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I agree. However, as there are 3 lines in the history, I guess that you should pass through a move request.
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Again, without a referred article explaining the meaning of these formulas, how there were obtained, and
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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than "quintic function". It's also more explicit. I'll move the article unless I hear objections. --
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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It is then a necessary (but not sufficient) condition that the irreducible solvable quintic
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Sorry, I'm probably writing in the wrong section, but my additions were removed (I'm new).
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As it is, it is a mathematical non-sense. I have tried to give it a meaning by replacing
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https://web.archive.org/web/20140701060849/http://library.wolfram.com/examples/quintic/
1456:{\displaystyle m={\frac {{b}^{3}\,c\,g-b\,{c}^{4}}{{b}^{2}\,{g}^{3}+{c}^{3}\,{g}^{2}}}}
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If you found an error with any archives or the URLs themselves, you can fix them with
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with rational coefficients must satisfy the simple quadratic curve
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Auxiliary equation is mentioned three times, it needs explaining!
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Please sign your posts in the talk pages with four tildes (~~~~).
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or of the rational root of the Cayley's resolvent in terms of
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K. S. UYANIK (53, Turkish citizen in Ankara city of TURKEY.)
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When you have finished reviewing my changes, please set the
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for additional information. I made the following changes:
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I have removed it. Thank you for pointing this out.—
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406:{\displaystyle y^{2}=(20-a)(5+a)\,}
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2281:Mid-priority mathematics articles
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120:and see a list of open tasks.
2276:C-Class mathematics articles
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1881:Hello fellow Wikipedians,
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1117:
1108:
1103:
1094:
1089:
1082:
1079:
1074:
1069:
1060:
1055:
1049:
1042:
1036:
1031:
1026:
1021:
1016:
1007:
1002:
996:
989:
983:
978:
973:
970:
963:
958:
952:
944:
939:
931:
926:
922:
908:
907:
906:
905:
904:
903:
887:
881:
875:
870:
865:
862:
855:
850:
844:
834:
829:
822:
816:
810:
805:
800:
797:
790:
785:
776:
771:
764:
761:
756:
751:
742:
737:
731:
724:
718:
713:
708:
705:
698:
693:
684:
679:
674:
669:
664:
655:
650:
644:
637:
631:
626:
621:
618:
611:
606:
600:
592:
587:
580:
574:
569:
565:
549:
548:
547:
546:
545:
544:
533:
530:
525:
521:
517:
514:
507:
503:
499:
494:
489:
472:
471:
445:
444:
432:
429:
426:
415:
414:
413:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
365:
361:
347:
346:
345:
332:
329:
324:
320:
316:
313:
310:
305:
301:
297:
294:
289:
285:
266:
263:
262:
261:
236:86.166.163.239
231:
228:
225:
224:
212:
209:
208:
203:
199:
197:
194:
193:
185:
184:
179:
173:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2293:
2282:
2279:
2277:
2274:
2273:
2271:
2264:
2263:
2259:
2255:
2250:
2248:
2247:Bring radical
2242:
2240:
2235:
2231:
2227:
2216:
2212:
2208:
2204:
2203:
2202:
2201:
2197:
2193:
2192:173.243.178.8
2185:
2181:
2177:
2173:
2168:
2161:
2153:
2152:
2151:
2149:
2145:
2141:
2128:
2124:
2120:
2115:
2112:
2111:
2110:
2109:
2105:
2101:
2097:
2089:
2083:
2079:
2075:
2071:
2067:
2063:
2062:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2033:
2032:
2031:
2030:
2026:
2022:
2018:
2014:
2006:
2004:
2003:
1998:
1993:
1992:
1981:
1977:
1974:
1970:
1969:
1968:
1961:
1955:
1951:
1947:
1943:
1937:
1932:
1928:
1922:
1918:
1914:
1907:
1903:
1899:
1898:
1897:
1895:
1891:
1887:
1882:
1876:
1870:
1866:
1862:
1857:
1856:
1855:
1854:
1853:
1852:
1848:
1844:
1835:
1834:
1833:
1831:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1798:
1797:
1796:
1792:
1788:
1787:104.132.34.86
1779:
1773:
1769:
1765:
1761:
1757:
1753:
1746:
1739:
1736:in terms of
1735:
1731:
1727:
1723:
1720:
1716:
1715:
1714:
1713:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1685:
1680:
1676:
1672:
1668:
1664:
1660:
1655:
1650:
1642:
1641:
1640:
1639:
1633:
1629:
1625:
1621:
1614:
1613:
1591:
1588:
1582:
1575:
1572:
1566:
1563:
1557:
1550:
1547:
1542:
1535:
1532:
1526:
1519:
1514:
1508:
1502:
1499:
1493:
1486:
1479:
1478:
1477:
1476:
1475:
1474:
1470:
1469:
1445:
1440:
1431:
1426:
1421:
1416:
1411:
1402:
1397:
1388:
1383:
1376:
1373:
1370:
1365:
1358:
1353:
1345:
1342:
1335:
1334:
1333:
1332:
1331:
1330:
1308:
1302:
1296:
1291:
1286:
1283:
1276:
1271:
1265:
1255:
1250:
1241:
1236:
1228:
1222:
1217:
1212:
1209:
1202:
1197:
1188:
1183:
1176:
1173:
1168:
1163:
1154:
1149:
1140:
1135:
1128:
1125:
1120:
1115:
1106:
1101:
1092:
1087:
1080:
1077:
1072:
1067:
1058:
1053:
1047:
1040:
1034:
1029:
1024:
1019:
1014:
1005:
1000:
994:
987:
981:
976:
971:
968:
961:
956:
950:
942:
937:
929:
924:
920:
912:
911:
910:
909:
885:
879:
873:
868:
863:
860:
853:
848:
842:
832:
827:
820:
814:
808:
803:
798:
795:
788:
783:
774:
769:
762:
759:
754:
749:
740:
735:
729:
722:
716:
711:
706:
703:
696:
691:
682:
677:
672:
667:
662:
653:
648:
642:
635:
629:
624:
619:
616:
609:
604:
598:
590:
585:
578:
572:
567:
563:
555:
554:
553:
552:
551:
550:
531:
528:
523:
519:
515:
512:
505:
501:
497:
492:
487:
478:
477:
476:
475:
474:
473:
469:
468:
467:
466:
462:
458:
454:
450:
430:
427:
424:
416:
395:
392:
389:
380:
377:
374:
368:
363:
359:
351:
350:
348:
330:
327:
322:
318:
314:
311:
308:
303:
299:
295:
292:
287:
283:
275:
274:
272:
271:
270:
264:
260:
256:
252:
248:
247:
246:
245:
241:
237:
229:
221:
216:
211:
210:
196:
195:
192:
191:
187:
186:
182:
177:
172:
171:
168:
153:
149:
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
34:
29:
28:
19:
2251:
2243:
2224:
2189:
2136:
2093:
2039:
2010:
1988:
1985:
1960:source check
1939:
1933:
1920:
1916:
1912:
1910:
1883:
1880:
1839:
1828:The section
1827:
1805:
1783:
1759:
1755:
1751:
1744:
1737:
1733:
1729:
1725:
1718:
1701:A.Samokrutov
1687:Please look
1663:WP:SECONDARY
1649:A.Samokrutov
1624:A.Samokrutov
1618:— Preceding
452:
448:
446:
268:
233:
214:
188:
180:
167:
148:Mid-priority
147:
107:
73:Mid‑priority
51:WikiProjects
2074:Anne Bauval
2021:Anne Bauval
1927:Sourcecheck
123:Mathematics
114:mathematics
70:Mathematics
2270:Categories
2167:resolvent.
1997:Report bug
2207:Anita5192
1980:this tool
1973:this tool
1471:solution:
2254:D.Lazard
2172:D.Lazard
2119:D.Lazard
2100:Macrakis
2052:D.Lazard
1986:Cheers.—
1861:D.Lazard
1810:D.Lazard
1764:D.Lazard
1671:D.Lazard
1632:contribs
1620:unsigned
457:D.Lazard
251:D.Lazard
215:365 days
181:Archives
2158:editor
1913:checked
1890:my edit
1689:quintic
1647:editor
150:on the
41:C-class
2239:WP:NOR
1921:failed
1843:Loraof
1697:septic
1693:sextic
1667:WP:COI
47:scale.
2040:Fixed
1806:Fixed
1659:WP:OR
2258:talk
2211:talk
2196:talk
2176:talk
2144:talk
2123:talk
2104:talk
2078:talk
2056:talk
2046:and
2025:talk
1917:true
1865:talk
1847:talk
1814:talk
1791:talk
1768:talk
1743:and
1705:talk
1675:talk
1628:talk
461:talk
255:talk
240:talk
1954:RfC
1931:).
1919:or
1904:to
451:by
142:Mid
2272::
2260:)
2213:)
2198:)
2178:)
2156:To
2146:)
2125:)
2106:)
2080:)
2072:.
2058:)
2027:)
2019:.
1967:.
1962:}}
1958:{{
1929:}}
1925:{{
1867:)
1849:)
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1808:.
1793:)
1785:--
1770:)
1762:.
1758:,
1754:,
1732:,
1728:,
1707:)
1695:,
1691:,
1677:)
1645:To
1630:•
1509:−
1374:−
1223:12
1177:22
1126:−
1081:22
1078:−
972:−
799:−
707:−
673:−
620:−
463:)
378:−
375:20
319:μ
300:μ
257:)
242:)
2256:(
2209:(
2194:(
2174:(
2162::
2142:(
2121:(
2102:(
2076:(
2054:(
2023:(
1999:)
1995:(
1982:.
1975:.
1863:(
1845:(
1812:(
1789:(
1766:(
1760:g
1756:c
1752:b
1748:2
1745:k
1741:1
1738:k
1734:g
1730:c
1726:b
1703:(
1673:(
1651::
1626:(
1592:5
1589:1
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1576:b
1573:+
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1564:2
1558:m
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1536:5
1533:3
1527:m
1520:g
1515:b
1503:5
1500:4
1494:m
1487:g
1446:2
1441:g
1432:3
1427:c
1422:+
1417:3
1412:g
1403:2
1398:b
1389:4
1384:c
1377:b
1371:g
1366:c
1359:3
1354:b
1346:=
1343:m
1309:4
1303:)
1297:3
1292:c
1287:+
1284:g
1277:2
1272:b
1266:(
1256:4
1251:g
1242:2
1237:c
1229:)
1218:c
1213:+
1210:g
1203:9
1198:c
1189:2
1184:b
1174:+
1169:2
1164:g
1155:6
1150:c
1141:4
1136:b
1129:6
1121:3
1116:g
1107:3
1102:c
1093:6
1088:b
1073:4
1068:g
1059:8
1054:b
1048:(
1041:)
1035:6
1030:c
1025:+
1020:2
1015:g
1006:4
1001:b
995:(
988:)
982:3
977:c
969:g
962:2
957:b
951:(
943:2
938:b
930:=
925:1
921:k
886:3
880:)
874:3
869:c
864:+
861:g
854:2
849:b
843:(
833:3
828:g
821:c
815:)
809:6
804:c
796:g
789:3
784:c
775:2
770:b
763:4
760:+
755:2
750:g
741:4
736:b
730:(
723:)
717:6
712:c
704:g
697:3
692:c
683:2
678:b
668:2
663:g
654:4
649:b
643:(
636:)
630:3
625:c
617:g
610:2
605:b
599:(
591:2
586:b
579:5
573:=
568:2
564:k
532:0
529:=
524:1
520:k
516:+
513:x
506:2
502:k
498:+
493:5
488:x
459:(
453:b
449:y
443:.
431:y
428:,
425:a
399:)
396:a
393:+
390:5
387:(
384:)
381:a
372:(
369:=
364:2
360:y
331:0
328:=
323:5
315:b
312:+
309:z
304:4
296:a
293:+
288:5
284:z
253:(
238:(
190:1
154:.
53::
20:)
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