3241:, who sent him a letter saying: "Your memoir on the general solution of equations is a work which I have always believed should be kept in mind by mathematicians and which, in my opinion, proves conclusively the algebraic unsolvability of general equations of higher than fourth degree." However, in general, Ruffini's proof was not considered convincing. Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini. But his memoir is so complicated that it is very difficult to determine the validity of his argument. It seems to me that his argument is not completely satisfying."
3219:
3802:. While Cauchy either did not notice Ruffini's assumption or felt that it was a minor one, most historians believe that the proof was not complete until Abel proved the theorem on natural irrationalities, which asserts that the assumption holds in the case of general polynomials. The Abel–Ruffini theorem is thus generally credited to Abel, who published a proof compressed into just six pages in 1824. (Abel adopted a very terse style to save paper and money: the proof was printed at his own expense.) A more elaborated version of the proof would be published in 1826.
3215:, he wrote "After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory." And he added "Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place." Actually, Gauss published nothing else on this subject.
3201:. This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions for equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusive proof. The first person who conjectured that the problem of solving quintics by radicals might be impossible to solve was
319:. With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100. Computing the solutions in radicals of solvable polynomials requires huge computations. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.
3817:
a memoir on his theory of solvability by radicals, which was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois was aware of the contributions of
Ruffini and Abel, since he wrote "It is a common truth,
3244:
The proof also, as it was discovered later, was incomplete. Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial. To
303:
might be soluble, with a special formula for each equation." However, the existence of specific equations that cannot be solved in radicals seems to be a consequence of Abel's proof, as the proof uses the fact that some polynomials in the coefficients are not the zero polynomial, and, given a finite
90:
refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of
3805:
Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals. Abel
2897:
that is a product of disjoint cycles of lengths 2 and 3 (in general, when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials, the degrees of the factors give the lengths of the disjoint cycles in some permutation belonging to the Galois group);
3835:
in 1845. When
Wantzel published it, he was already aware of the contributions by Galois and he mentions that, whereas Abel's proof is valid only for general polynomials, Galois' approach can be used to provide a concrete polynomial of degree 5 whose roots cannot be expressed in radicals from its
298:
Abel's proof of the theorem does not explicitly contain the assertion that there are specific equations that cannot be solved by radicals. Such an assertion is not a consequence of Abel's statement of the theorem, as the statement does not exclude the possibility that "every particular
247:
The Abel–Ruffini theorem proves that this is impossible. However, this impossibility does not imply that a specific equation of any degree cannot be solved in radicals. On the contrary, there are equations of any degree that can be solved in radicals. This is the case of the equation
212:
From the 16th century to beginning of the 19th century, the main problem of algebra was to search for a formula for the solutions of polynomial equations of degree five and higher, hence the name the "fundamental theorem of algebra". This meant a
2068:
2565:
as a generic equation. This finishes the proof that the Galois group of a general equation is the symmetric group, and thus proves the original Abel–Ruffini theorem, which asserts that the general polynomial equation of degree
939:
Conversely, if one has such a sequence of fields, the equation is solvable in terms of radicals. For proving this, it suffices to prove that a normal extension with a cyclic Galois group can be built from a succession of
331:. However, Galois theory allows a better understanding of the subject, and modern proofs are generally based on it, while the original proofs of the Abel–Ruffini theorem are still presented for historical purposes.
3813:, "The proofs of Ruffini and of Abel were soon superseded by the crowning achievement of this line of research: Galois' discoveries in the theory of equations." In 1830, Galois (at the age of 18) submitted to the
451:
3830:
accompanied by some of his own explanations. Prior to this publication, Liouville announced Galois' result to the academy in a speech he gave on 4 July 1843. A simplification of Abel's proof was published by
1583:
927:
So, if one starts from a solution in terms of radicals, one gets an increasing sequence of fields such that the last one contains the solution, and each is a normal extension of the preceding one with a
374:. Such an expression may be viewed as the description of a computation that starts from the coefficients of the equation to be solved and proceeds by computing some numbers, one after the other.
3211:(which would be published only in 1801) that "there is little doubt that this problem does not so much defy modern methods of analysis as that it proposes the impossible". The next year, in his
2246:
2134:
1702:
3822:
cannot be solved by radicals... this truth has become common (by hearsay) despite the fact that geometers have ignored the proofs of Abel and
Ruffini..." Galois then died in 1832 and his paper
3800:
3685:
2833:
2395:
1125:
1069:
655:
1026:
3079:
2183:
556:
2429:
1900:
1854:
1752:
1632:
3110:
3038:
2723:
1805:
1400:
1369:
1330:
1284:
1246:
3300:
2999:
2871:
597:
494:
923:
4487:
3936:
3707:
2961:
2753:
3155:
2634:
688:
141:
3476:
3438:
3400:
3511:
2680:
285:
3736:
846:
755:
2563:
2355:
3592:
3565:
3538:
3362:
3333:
2927:
2520:
2493:
2462:
2309:
2278:
1197:
877:
813:
782:
715:
3621:
990:
194:
for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.
2895:
1908:
3193:
began the groundwork that unified the many different methods that had been used up to that point to solve equations, relating them to the theory of groups of
2434:
1163:
3959:
1413:. Thus, the Abel–Ruffini theorem results from the existence of polynomials with a symmetric Galois group; this will be shown in the next section.
4345:
3177:
quintic is solvable in radicals if and only, when its coefficients are substituted in Cayley's resolvent, the resulting sextic polynomial has a
4638:
4515:
4460:
4425:
4394:
4291:
4273:
Teoria generale delle equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto
4184:
4051:
3877:
397:
4674:
3218:
3999:
4547:
4223:
3960:"Mémoire sur les équations algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré"
381:
that contains all numbers that have been computed so far. This field is changed only for the steps involving the computation of an
4622:
4539:
1482:
315:
that allows deciding, for any given equation, whether it is solvable in radicals. This was purely theoretical before the rise of
690:
An algebraic solution of the initial polynomial equation exists if and only if there exists such a sequence of fields such that
4479:
202:
4364:
4018:
3978:
3237:
in 1799. He sent his proof to several mathematicians to get it acknowledged, amongst them
Lagrange (who did not reply) and
4115:
4110:
3234:
77:
2188:
2076:
1644:
4378:
4035:
3207:
2930:
4000:"Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré"
3741:
3626:
1138:
The preceding section shows that an equation is solvable in terms of radicals if and only if the Galois group of its
201:, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the
1765:, this equation is not solvable in radicals. In view of the preceding sections, this results from the fact that the
4664:
4583:
4352:
4006:
3966:
335:
334:
The proofs based on Galois theory comprise four main steps: the characterization of solvable equations in terms of
3938:
Riflessioni intorno alla soluzione delle equazioni algebraiche generali opuscolo del cav. dott. Paolo
Ruffini ...
3848:
3814:
957:
354:
is not solvable if its degree is five or higher; and the existence of polynomials with a symmetric Galois group.
2758:
304:
number of polynomials, there are values of the variables at which none of the polynomials takes the value zero.
4669:
2360:
1635:
1090:
1034:
363:
218:
70:
602:
999:
849:
80:, who made an incomplete proof in 1799 (which was refined and completed in 1813 and accepted by Cauchy) and
155:
The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the
4659:
3174:
3047:
2465:
2139:
499:
234:
2400:
1859:
1825:
1711:
1591:
4246:
3238:
3190:
3084:
3012:
2697:
1779:
1374:
1343:
1304:
1258:
1220:
953:
367:
339:
292:
3248:
3173:
of degree six whose coefficients are polynomials in the coefficients of a generic quintic. A specific
2966:
2838:
4250:
3202:
1444:
214:
50:
561:
459:
3198:
2316:
1639:
1440:
882:
378:
316:
238:
175:
3690:
2936:
2728:
4511:"Démonstration de l'impossibilité de résoudre toutes les équations algébriques avec des radicaux"
4324:
4308:
4168:
4132:
4075:
3910:
3894:
3115:
3006:
2594:
2324:
1808:
1072:
660:
101:
54:
4475:
3443:
3405:
3367:
308:
206:
4066:
Fieker, Claus; Klüners, Jürgen (2014), "Computation of Galois groups of rational polynomials",
3481:
2646:
251:
4634:
4614:
4543:
4456:
4421:
4390:
4341:
4219:
4180:
4047:
3995:
3955:
3712:
1705:
1424:
1253:
941:
724:, which are fundamental for the theory, one must refine the sequence of fields as follows. If
179:
156:
81:
31:
1819:, where the splitting field is the smallest field containing all the roots of the equation).
1423:, the symmetric group and all its subgroups are solvable. This explains the existence of the
818:
727:
4626:
4596:
4561:
4431:
4417:
4300:
4229:
4190:
4176:
4148:
4124:
4085:
3918:
3886:
3827:
3166:
2533:
2330:
721:
300:
58:
4557:
4320:
4144:
4097:
3906:
3570:
3543:
3516:
3338:
3309:
2905:
2498:
2471:
2440:
2287:
2251:
1175:
855:
791:
760:
693:
4565:
4553:
4448:
4435:
4316:
4233:
4194:
4152:
4140:
4093:
3922:
3902:
3840:
3810:
3597:
3178:
3041:
1812:
1774:
1755:
1432:
1291:
1287:
1215:
1170:
1147:
1139:
371:
351:
191:
183:
164:
4536:
Abel's
Theorem in Problems and Solutions: Based on the Lectures of Professor V. I. Arnold
3306:, one of its roots (all of them, actually) can be expressed as the sum of a cube root of
2063:{\displaystyle P(x)=x^{n}+b_{1}x^{n-1}+\cdots +b_{n-1}x+b_{n}=(x-x_{1})\cdots (x-x_{n}).}
992:
and the subgroups of the Galois group of the extension. This correspondence maps a field
967:
91:
the equation are not the zero polynomial. This improved statement follows directly from
4506:
3832:
3303:
2880:
1403:
1151:
1143:
785:
347:
230:
4289:
Kiernan, B. Melvin (1971), "The
Development of Galois Theory from Lagrange to Artin",
4653:
4619:
Topological Galois Theory: Solvability and
Unsolvability of Equations in Finite Terms
4579:"Short Proof of Abel's Theorem that 5th Degree Polynomial Equations Cannot be Solved"
4328:
4268:
3914:
3222:
2586:
1436:
1428:
1337:
1159:
328:
312:
187:
160:
92:
17:
4510:
3245:
see why this is really an extra assumption, consider, for instance, the polynomial
1766:
1448:
1333:
1155:
1029:
961:
933:
929:
343:
197:
The fact that every polynomial equation of positive degree has solutions, possibly
62:
3194:
3165:
Testing whether a specific quintic is solvable in radicals can be done by using
2281:
226:
205:, which does not provide any tool for computing exactly the solutions, although
198:
38:
4212:
Abel's Proof: An Essay on the
Sources and Meaning of Mathematical Unsolvability
2357:
and is thus fixed by all these automorphisms. It follows that the Galois group
4630:
4577:
4356:
4089:
4010:
3970:
3170:
1199:
is not solvable, and that there are polynomials with symmetric Galois groups.
148:
4386:
4215:
4043:
1451:
is solvable (the term "solvable group" takes its origin from this theorem).
3875:
Ayoub, Raymond G. (1980), "Paolo
Ruffini's Contributions to the Quintic",
2686:
be its Galois group, which acts faithfully on the set of complex roots of
1902:
be new indeterminates, aimed to be the roots, and consider the polynomial
69:
means that the coefficients of the equation are viewed and manipulated as
3844:
382:
222:
4312:
3898:
4480:"Mémoire sur les conditions de resolubilité des équations par radicaux"
4304:
4136:
3890:
221:
involving only the coefficients of the equation, and the operations of
1169:
So, for proving the Abel–Ruffini theorem, it remains to show that the
4249:(1869) , "Réflexions sur la résolution algébrique des équations", in
3824:
Mémoire sur les conditions de resolubilité des équations par radicaux
3212:
146:
is the simplest equation that cannot be solved in radicals, and that
4128:
446:{\displaystyle F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{k}}
1336:(that is, it does not have any nontrivial normal subgroup) and not
152:
polynomials of degree five or higher cannot be solved in radicals.
4080:
3847:
of the Abel–Ruffini theorem, which served as a starting point for
3217:
1758:
coefficients. The original Abel–Ruffini theorem asserts that, for
3806:
was working on a complete characterization when he died in 1829.
1146:, that is, it contains a sequence of subgroups such that each is
4113:(1995), "Niels Hendrik Abel and Equations of the Fifth Degree",
377:
At each step of the computation, one may consider the smallest
4598:
Arnold's Elementary Proof of the Insolvability of the Quintic
2944:
2736:
3091:
3060:
3019:
2704:
2407:
1832:
1786:
1578:{\displaystyle x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}=0,}
1381:
1350:
1311:
1265:
1227:
342:
between subfields of a given field and the subgroups of its
209:
allows approximating the solutions to any desired accuracy.
27:
Equations of degree 5 or higher cannot be solved by radicals
370:(addition, subtraction, multiplication, and division), and
3826:
remained unpublished until 1846, when it was published by
2636:
is not solvable in radicals, as will be explained below.
3818:
today, that the general equation of degree greater than
3009:. Because 5 is prime, any transposition and 5-cycle in
295:, all of whose solutions can be expressed in radicals.
3744:
3715:
3693:
3629:
3600:
3573:
3546:
3519:
3484:
3446:
3408:
3370:
3341:
3312:
3251:
3118:
3087:
3050:
3015:
2969:
2939:
2908:
2883:
2841:
2761:
2731:
2700:
2649:
2597:
2536:
2501:
2474:
2443:
2403:
2363:
2333:
2290:
2254:
2191:
2142:
2079:
1911:
1862:
1828:
1782:
1714:
1647:
1594:
1485:
1377:
1346:
1307:
1261:
1223:
1178:
1093:
1037:
1002:
970:
885:
858:
821:
794:
763:
730:
696:
663:
605:
564:
502:
462:
400:
362:
An algebraic solution of a polynomial equation is an
254:
104:
3205:, who wrote in 1798 in section 359 of his book
1142:(the smallest field that contains all the roots) is
4257:, vol. III, Gauthier-Villars, pp. 205–421
2587:
Galois theory § A non-solvable quintic example
2241:{\displaystyle K=\mathbb {Q} (b_{1},\ldots ,b_{n})}
2129:{\displaystyle H=\mathbb {Q} (x_{1},\ldots ,x_{n})}
1697:{\displaystyle F=\mathbb {Q} (a_{1},\ldots ,a_{n})}
93:
Galois theory § A non-solvable quintic example
3794:
3730:
3701:
3679:
3615:
3586:
3559:
3532:
3505:
3470:
3432:
3394:
3356:
3327:
3294:
3149:
3104:
3073:
3032:
2993:
2955:
2921:
2889:
2865:
2827:
2747:
2717:
2674:
2628:
2557:
2514:
2487:
2456:
2423:
2389:
2349:
2303:
2272:
2240:
2177:
2128:
2062:
1894:
1848:
1799:
1746:
1696:
1626:
1577:
1394:
1363:
1324:
1278:
1240:
1191:
1119:
1063:
1020:
984:
917:
871:
840:
807:
776:
749:
709:
682:
649:
591:
550:
488:
445:
279:
135:
4171:(2016), "Ruffini and Abel on General Equations",
2248:be its subfield generated by the coefficients of
346:for expressing this characterization in terms of
1166:shows that the two definitions are equivalent).
167:for degrees two, three, and four, respectively.
3795:{\displaystyle \mathbf {Q} (r_{1},r_{2},r_{3})}
3680:{\displaystyle \mathbf {Q} (r_{1},r_{2},r_{3})}
3042:Symmetric group § Generators and relations
327:The proof of the Abel–Ruffini theorem predates
4275:(in Italian), Stamperia di S. Tommaso d'Aquino
391:So, an algebraic solution produces a sequence
4407:
4405:
4163:
4161:
1158:. (Solvable groups are commonly defined with
8:
4488:Journal de Mathématiques Pures et Appliquées
4346:"Sur la resolution algébrique des équations"
3941:(in Italian). presso la Societa Tipografica.
2435:fundamental theorem of symmetric polynomials
1164:fundamental theorem of finite abelian groups
307:Soon after Abel's publication of its proof,
1856:it is simpler to start from the roots. Let
1162:instead of cyclic quotient groups, but the
4363:(in French), vol. II (2nd ed.),
4284:
4282:
4068:LMS Journal of Computation and Mathematics
2828:{\displaystyle (x^{2}+x+1)(x^{3}+x^{2}+1)}
2136:be the field of the rational fractions in
4381:(2015), "The Idea Behind Galois Theory",
4079:
4030:
4028:
4017:(in French), vol. I (2nd ed.),
3977:(in French), vol. I (2nd ed.),
3783:
3770:
3757:
3745:
3743:
3714:
3694:
3692:
3668:
3655:
3642:
3630:
3628:
3599:
3578:
3572:
3551:
3545:
3524:
3518:
3483:
3445:
3407:
3369:
3340:
3311:
3271:
3250:
3123:
3117:
3096:
3090:
3089:
3086:
3065:
3059:
3058:
3049:
3024:
3018:
3017:
3014:
2976:
2972:
2971:
2968:
2947:
2943:
2938:
2913:
2907:
2882:
2848:
2844:
2843:
2840:
2810:
2797:
2769:
2760:
2739:
2735:
2730:
2709:
2703:
2702:
2699:
2654:
2648:
2602:
2596:
2535:
2506:
2500:
2479:
2473:
2448:
2442:
2412:
2406:
2405:
2402:
2390:{\displaystyle \operatorname {Gal} (H/K)}
2376:
2362:
2338:
2332:
2295:
2289:
2253:
2229:
2210:
2199:
2198:
2190:
2166:
2147:
2141:
2117:
2098:
2087:
2086:
2078:
2048:
2023:
2001:
1979:
1954:
1944:
1931:
1910:
1886:
1867:
1861:
1837:
1831:
1830:
1827:
1815:of the equation that fix the elements of
1791:
1785:
1784:
1781:
1738:
1719:
1713:
1685:
1666:
1655:
1654:
1646:
1618:
1599:
1593:
1560:
1538:
1513:
1503:
1490:
1484:
1386:
1380:
1379:
1376:
1355:
1349:
1348:
1345:
1316:
1310:
1309:
1306:
1270:
1264:
1263:
1260:
1232:
1226:
1225:
1222:
1183:
1177:
1120:{\displaystyle \operatorname {Gal} (F/E)}
1106:
1092:
1064:{\displaystyle \operatorname {Gal} (F/K)}
1050:
1036:
1001:
974:
969:
903:
890:
884:
863:
857:
826:
820:
799:
793:
768:
762:
735:
729:
701:
695:
668:
662:
635:
620:
615:
610:
604:
563:
539:
520:
507:
501:
480:
467:
461:
437:
418:
405:
399:
259:
253:
109:
103:
2690:. Numbering the roots lets one identify
2468:, and thus that the map that sends each
1455:Polynomials with symmetric Galois groups
1083:fixed, and, conversely, maps a subgroup
650:{\displaystyle x_{i}^{n_{i}}\in F_{i-1}}
4205:
4203:
3990:
3988:
3950:
3948:
3870:
3868:
3866:
3864:
3860:
3233:The theorem was first nearly proved by
2694:with a subgroup of the symmetric group
1807:(this Galois group is the group of the
1638:. This is an equation defined over the
1292:Symmetric group § Normal subgroups
1021:{\displaystyle E\subseteq K\subseteq F}
4621:, Springer Monographs in Mathematics,
4451:(2009), "Galois Theory of Equations",
4412:Tignol, Jean-Pierre (2016), "Galois",
4455:, vol. 1 (2nd ed.), Dover,
4414:Galois' Theory of Algebraic Equations
4361:Œuvres Complètes de Niels Henrik Abel
4292:Archive for History of Exact Sciences
4173:Galois' Theory of Algebraic Equations
4015:Œuvres Complètes de Niels Henrik Abel
3975:Œuvres Complètes de Niels Henrik Abel
3878:Archive for History of Exact Sciences
3623:are all real and therefore the field
1822:For proving that the Galois group is
178:of degree two can be solved with the
7:
3074:{\displaystyle G={\mathcal {S}}_{5}}
2178:{\displaystyle x_{1},\ldots ,x_{n},}
551:{\displaystyle F_{i}=F_{i-1}(x_{i})}
358:Algebraic solutions and field theory
4038:(2015), "Historical Introduction",
2530:. This means that one may consider
2424:{\displaystyle {\mathcal {S}}_{n}.}
1895:{\displaystyle x_{1},\ldots ,x_{n}}
1849:{\displaystyle {\mathcal {S}}_{n},}
1747:{\displaystyle a_{1},\ldots ,a_{n}}
1627:{\displaystyle a_{1},\ldots ,a_{n}}
95:. Galois theory implies also that
4516:Nouvelles Annales de Mathématiques
3105:{\displaystyle {\mathcal {S}}_{5}}
3033:{\displaystyle {\mathcal {S}}_{5}}
2718:{\displaystyle {\mathcal {S}}_{5}}
1800:{\displaystyle {\mathcal {S}}_{n}}
1435:formulas, since a major result of
1395:{\displaystyle {\mathcal {S}}_{n}}
1364:{\displaystyle {\mathcal {A}}_{n}}
1325:{\displaystyle {\mathcal {A}}_{n}}
1279:{\displaystyle {\mathcal {A}}_{n}}
1241:{\displaystyle {\mathcal {S}}_{n}}
25:
3295:{\displaystyle P(x)=x^{3}-15x-20}
2570:cannot be solved in radicals for
3746:
3695:
3631:
3001:, the same principle shows that
2994:{\displaystyle \mathbb {F} _{3}}
2866:{\displaystyle \mathbb {F} _{2}}
1127:to the field of the elements of
311:introduced a theory, now called
84:, who provided a proof in 1824.
3227:Teoria generale delle equazioni
291:, and the equations defined by
4534:Alekseev, Valeriy B. (2004),
3789:
3750:
3674:
3635:
3610:
3604:
3494:
3488:
3459:
3450:
3421:
3412:
3383:
3374:
3261:
3255:
3112:is not solvable, the equation
3040:generate the whole group; see
2988:
2982:
2860:
2854:
2822:
2790:
2787:
2762:
2546:
2540:
2384:
2370:
2264:
2258:
2235:
2203:
2123:
2091:
2054:
2035:
2029:
2010:
1921:
1915:
1691:
1659:
1472:polynomial equation of degree
1114:
1100:
1058:
1044:
909:
896:
592:{\displaystyle i=1,\ldots ,k,}
545:
532:
489:{\displaystyle x_{i}\in F_{i}}
203:fundamental theorem of algebra
1:
4116:American Mathematical Monthly
3157:is not solvable in radicals.
1150:in the preceding one, with a
918:{\displaystyle K_{i}(x_{i}).}
182:, which has been known since
3702:{\displaystyle \mathbf {R} }
2956:{\displaystyle q{\bmod {3}}}
2748:{\displaystyle q{\bmod {2}}}
2522:is a field isomorphism from
2319:imply that every element of
964:of a normal field extension
47:Abel's impossibility theorem
3364:. On the other hand, since
3208:Disquisitiones Arithmeticae
3150:{\displaystyle x^{5}-x-1=0}
2629:{\displaystyle x^{5}-x-1=0}
788:, one introduces the field
683:{\displaystyle n_{i}>1.}
136:{\displaystyle x^{5}-x-1=0}
76:The theorem is named after
4691:
4675:Theorems about polynomials
4540:Kluwer Academic Publishers
3471:{\displaystyle P(-1)<0}
3433:{\displaystyle P(-2)>0}
3395:{\displaystyle P(-3)<0}
2584:
190:for degree three, and the
49:) states that there is no
29:
4631:10.1007/978-3-642-38871-2
4090:10.1112/S1461157013000302
3849:topological Galois theory
3815:Paris Academy of Sciences
3506:{\displaystyle P(5)>0}
3160:
2675:{\displaystyle x^{5}-x-1}
1340:. This implies that both
1203:Solvable symmetric groups
958:one to one correspondence
366:involving the four basic
280:{\displaystyle x^{n}-1=0}
61:or higher with arbitrary
3731:{\displaystyle 10\pm 5i}
2311:induce automorphisms of
1301:, the alternating group
456:of fields, and elements
30:Not to be confused with
3935:Ruffini, Paolo (1813).
3709:. But then the numbers
2877:contains a permutation
2397:is the symmetric group
1773:of the equation is the
1416:On the other hand, for
850:primitive root of unity
841:{\displaystyle F_{i-1}}
750:{\displaystyle F_{i-1}}
4247:Lagrange, Joseph-Louis
3796:
3732:
3703:
3681:
3617:
3588:
3561:
3534:
3507:
3472:
3434:
3396:
3358:
3329:
3296:
3230:
3151:
3106:
3075:
3034:
2995:
2957:
2923:
2891:
2867:
2829:
2749:
2719:
2676:
2630:
2559:
2558:{\displaystyle P(x)=0}
2516:
2489:
2458:
2425:
2391:
2351:
2350:{\displaystyle x_{i},}
2305:
2274:
2242:
2179:
2130:
2064:
1896:
1850:
1801:
1748:
1698:
1628:
1579:
1396:
1365:
1326:
1280:
1242:
1193:
1121:
1065:
1022:
986:
919:
873:
842:
809:
778:
751:
711:
684:
651:
593:
552:
490:
447:
293:cyclotomic polynomials
281:
137:
4251:Serret, Joseph-Alfred
4210:Pesic, Peter (2004),
3797:
3733:
3704:
3682:
3618:
3589:
3587:{\displaystyle r_{3}}
3562:
3560:{\displaystyle r_{2}}
3535:
3533:{\displaystyle r_{1}}
3508:
3473:
3435:
3397:
3359:
3357:{\displaystyle 10-5i}
3330:
3328:{\displaystyle 10+5i}
3297:
3239:Augustin-Louis Cauchy
3221:
3191:Joseph Louis Lagrange
3171:univariate polynomial
3152:
3107:
3076:
3035:
2996:
2958:
2924:
2922:{\displaystyle g^{3}}
2892:
2868:
2830:
2750:
2720:
2677:
2631:
2560:
2517:
2515:{\displaystyle b_{i}}
2495:to the corresponding
2490:
2488:{\displaystyle a_{i}}
2466:algebraic independent
2459:
2457:{\displaystyle b_{i}}
2426:
2392:
2352:
2306:
2304:{\displaystyle x_{i}}
2275:
2273:{\displaystyle P(x).}
2243:
2180:
2131:
2065:
1897:
1851:
1802:
1749:
1699:
1629:
1580:
1397:
1366:
1327:
1281:
1243:
1194:
1192:{\displaystyle S_{5}}
1122:
1066:
1023:
987:
954:Galois correspondence
948:Galois correspondence
920:
874:
872:{\displaystyle F_{i}}
843:
810:
808:{\displaystyle K_{i}}
779:
777:{\displaystyle n_{i}}
757:does not contain all
752:
717:contains a solution.
712:
710:{\displaystyle F_{k}}
685:
652:
594:
553:
491:
448:
368:arithmetic operations
350:; the proof that the
340:Galois correspondence
282:
138:
3742:
3713:
3691:
3627:
3616:{\displaystyle P(x)}
3598:
3571:
3544:
3517:
3482:
3444:
3406:
3368:
3339:
3335:with a cube root of
3310:
3249:
3203:Carl Friedrich Gauss
3116:
3085:
3048:
3013:
2967:
2937:
2906:
2881:
2839:
2759:
2729:
2698:
2647:
2595:
2534:
2499:
2472:
2441:
2401:
2361:
2331:
2288:
2252:
2189:
2140:
2077:
1909:
1860:
1826:
1780:
1712:
1645:
1592:
1483:
1445:solution in radicals
1375:
1344:
1305:
1259:
1221:
1176:
1091:
1035:
1000:
968:
883:
856:
852:, and one redefines
819:
792:
761:
728:
694:
661:
603:
562:
500:
460:
398:
317:electronic computers
252:
215:solution in radicals
176:Polynomial equations
102:
88:Abel–Ruffini theorem
55:polynomial equations
51:solution in radicals
43:Abel–Ruffini theorem
18:Abel-Ruffini theorem
4169:Tignol, Jean-Pierre
3199:Lagrange resolvents
1809:field automorphisms
1447:if and only if its
1441:polynomial equation
985:{\displaystyle F/E}
627:
4615:Khovanskii, Askold
4367:, pp. 217–243
4365:Grøndahl & Søn
4342:Abel, Niels Henrik
4305:10.1007/BF00327219
4255:Œuvres de Lagrange
4019:Grøndahl & Søn
3996:Abel, Niels Henrik
3979:Grøndahl & Søn
3956:Abel, Niels Henrik
3891:10.1007/BF00357046
3792:
3728:
3699:
3677:
3613:
3584:
3557:
3530:
3503:
3468:
3430:
3392:
3354:
3325:
3292:
3231:
3167:Cayley's resolvent
3161:Cayley's resolvent
3147:
3102:
3081:. Since the group
3071:
3030:
2991:
2963:is irreducible in
2953:
2919:
2887:
2863:
2825:
2745:
2715:
2672:
2626:
2555:
2512:
2485:
2454:
2421:
2387:
2347:
2325:symmetric function
2301:
2270:
2238:
2175:
2126:
2060:
1892:
1846:
1797:
1744:
1706:rational fractions
1694:
1624:
1575:
1392:
1361:
1322:
1276:
1238:
1189:
1131:that are fixed by
1117:
1061:
1018:
982:
942:radical extensions
915:
869:
838:
805:
774:
747:
707:
680:
647:
606:
589:
548:
486:
443:
277:
242:th root extraction
133:
4665:Niels Henrik Abel
4640:978-3-642-38870-5
4595:Goldmakher, Leo,
4462:978-0-486-47189-1
4427:978-981-4704-69-4
4396:978-1-4822-4582-0
4186:978-981-4704-69-4
4111:Rosen, Michael I.
4053:978-1-4822-4582-0
3845:topological proof
3738:cannot belong to
3687:is a subfield of
3304:Cardano's formula
3197:, in the form of
2890:{\displaystyle g}
2437:implies that the
1254:alternating group
722:normal extensions
657:for some integer
338:; the use of the
180:quadratic formula
157:quadratic formula
82:Niels Henrik Abel
16:(Redirected from
4682:
4644:
4643:
4611:
4605:
4604:
4603:
4592:
4586:
4580:
4575:
4569:
4568:
4531:
4525:
4524:
4503:
4497:
4496:
4484:
4476:Galois, Évariste
4472:
4466:
4465:
4449:Jacobson, Nathan
4445:
4439:
4438:
4418:World Scientific
4416:(2nd ed.),
4409:
4400:
4399:
4385:(4th ed.),
4375:
4369:
4368:
4350:
4338:
4332:
4331:
4286:
4277:
4276:
4265:
4259:
4258:
4243:
4237:
4236:
4207:
4198:
4197:
4177:World Scientific
4175:(2nd ed.),
4165:
4156:
4155:
4107:
4101:
4100:
4083:
4063:
4057:
4056:
4042:(4th ed.),
4032:
4023:
4022:
4021:, pp. 66–87
4004:
3992:
3983:
3982:
3981:, pp. 28–33
3964:
3952:
3943:
3942:
3932:
3926:
3925:
3872:
3828:Joseph Liouville
3821:
3801:
3799:
3798:
3793:
3788:
3787:
3775:
3774:
3762:
3761:
3749:
3737:
3735:
3734:
3729:
3708:
3706:
3705:
3700:
3698:
3686:
3684:
3683:
3678:
3673:
3672:
3660:
3659:
3647:
3646:
3634:
3622:
3620:
3619:
3614:
3593:
3591:
3590:
3585:
3583:
3582:
3566:
3564:
3563:
3558:
3556:
3555:
3539:
3537:
3536:
3531:
3529:
3528:
3512:
3510:
3509:
3504:
3477:
3475:
3474:
3469:
3439:
3437:
3436:
3431:
3401:
3399:
3398:
3393:
3363:
3361:
3360:
3355:
3334:
3332:
3331:
3326:
3301:
3299:
3298:
3293:
3276:
3275:
3156:
3154:
3153:
3148:
3128:
3127:
3111:
3109:
3108:
3103:
3101:
3100:
3095:
3094:
3080:
3078:
3077:
3072:
3070:
3069:
3064:
3063:
3039:
3037:
3036:
3031:
3029:
3028:
3023:
3022:
3004:
3000:
2998:
2997:
2992:
2981:
2980:
2975:
2962:
2960:
2959:
2954:
2952:
2951:
2928:
2926:
2925:
2920:
2918:
2917:
2901:
2896:
2894:
2893:
2888:
2876:
2872:
2870:
2869:
2864:
2853:
2852:
2847:
2834:
2832:
2831:
2826:
2815:
2814:
2802:
2801:
2774:
2773:
2754:
2752:
2751:
2746:
2744:
2743:
2724:
2722:
2721:
2716:
2714:
2713:
2708:
2707:
2693:
2689:
2685:
2681:
2679:
2678:
2673:
2659:
2658:
2642:
2635:
2633:
2632:
2627:
2607:
2606:
2581:Explicit example
2576:
2569:
2564:
2562:
2561:
2556:
2529:
2525:
2521:
2519:
2518:
2513:
2511:
2510:
2494:
2492:
2491:
2486:
2484:
2483:
2463:
2461:
2460:
2455:
2453:
2452:
2430:
2428:
2427:
2422:
2417:
2416:
2411:
2410:
2396:
2394:
2393:
2388:
2380:
2356:
2354:
2353:
2348:
2343:
2342:
2322:
2317:Vieta's formulas
2314:
2310:
2308:
2307:
2302:
2300:
2299:
2279:
2277:
2276:
2271:
2247:
2245:
2244:
2239:
2234:
2233:
2215:
2214:
2202:
2184:
2182:
2181:
2176:
2171:
2170:
2152:
2151:
2135:
2133:
2132:
2127:
2122:
2121:
2103:
2102:
2090:
2069:
2067:
2066:
2061:
2053:
2052:
2028:
2027:
2006:
2005:
1990:
1989:
1965:
1964:
1949:
1948:
1936:
1935:
1901:
1899:
1898:
1893:
1891:
1890:
1872:
1871:
1855:
1853:
1852:
1847:
1842:
1841:
1836:
1835:
1818:
1806:
1804:
1803:
1798:
1796:
1795:
1790:
1789:
1772:
1764:
1753:
1751:
1750:
1745:
1743:
1742:
1724:
1723:
1703:
1701:
1700:
1695:
1690:
1689:
1671:
1670:
1658:
1633:
1631:
1630:
1625:
1623:
1622:
1604:
1603:
1584:
1582:
1581:
1576:
1565:
1564:
1549:
1548:
1524:
1523:
1508:
1507:
1495:
1494:
1476:is the equation
1475:
1460:General equation
1422:
1412:
1401:
1399:
1398:
1393:
1391:
1390:
1385:
1384:
1370:
1368:
1367:
1362:
1360:
1359:
1354:
1353:
1331:
1329:
1328:
1323:
1321:
1320:
1315:
1314:
1300:
1286:as a nontrivial
1285:
1283:
1282:
1277:
1275:
1274:
1269:
1268:
1251:
1247:
1245:
1244:
1239:
1237:
1236:
1231:
1230:
1213:
1198:
1196:
1195:
1190:
1188:
1187:
1134:
1130:
1126:
1124:
1123:
1118:
1110:
1086:
1082:
1078:
1070:
1068:
1067:
1062:
1054:
1027:
1025:
1024:
1019:
995:
991:
989:
988:
983:
978:
924:
922:
921:
916:
908:
907:
895:
894:
878:
876:
875:
870:
868:
867:
847:
845:
844:
839:
837:
836:
814:
812:
811:
806:
804:
803:
783:
781:
780:
775:
773:
772:
756:
754:
753:
748:
746:
745:
716:
714:
713:
708:
706:
705:
689:
687:
686:
681:
673:
672:
656:
654:
653:
648:
646:
645:
626:
625:
624:
614:
598:
596:
595:
590:
557:
555:
554:
549:
544:
543:
531:
530:
512:
511:
495:
493:
492:
487:
485:
484:
472:
471:
452:
450:
449:
444:
442:
441:
423:
422:
410:
409:
385:
372:root extractions
301:quintic equation
290:
286:
284:
283:
278:
264:
263:
241:
186:. Similarly the
142:
140:
139:
134:
114:
113:
21:
4690:
4689:
4685:
4684:
4683:
4681:
4680:
4679:
4670:Solvable groups
4650:
4649:
4648:
4647:
4641:
4623:Springer-Verlag
4613:
4612:
4608:
4601:
4594:
4593:
4589:
4578:
4576:
4572:
4550:
4533:
4532:
4528:
4507:Wantzel, Pierre
4505:
4504:
4500:
4482:
4474:
4473:
4469:
4463:
4447:
4446:
4442:
4428:
4411:
4410:
4403:
4397:
4377:
4376:
4372:
4348:
4340:
4339:
4335:
4299:(1/2): 40–154,
4288:
4287:
4280:
4267:
4266:
4262:
4245:
4244:
4240:
4226:
4209:
4208:
4201:
4187:
4167:
4166:
4159:
4129:10.2307/2974763
4109:
4108:
4104:
4065:
4064:
4060:
4054:
4034:
4033:
4026:
4002:
3994:
3993:
3986:
3962:
3954:
3953:
3946:
3934:
3933:
3929:
3874:
3873:
3862:
3857:
3841:Vladimir Arnold
3819:
3811:Nathan Jacobson
3779:
3766:
3753:
3740:
3739:
3711:
3710:
3689:
3688:
3664:
3651:
3638:
3625:
3624:
3596:
3595:
3574:
3569:
3568:
3547:
3542:
3541:
3520:
3515:
3514:
3480:
3479:
3442:
3441:
3404:
3403:
3366:
3365:
3337:
3336:
3308:
3307:
3302:. According to
3267:
3247:
3246:
3187:
3163:
3119:
3114:
3113:
3088:
3083:
3082:
3057:
3046:
3045:
3016:
3011:
3010:
3002:
2970:
2965:
2964:
2935:
2934:
2909:
2904:
2903:
2899:
2879:
2878:
2874:
2842:
2837:
2836:
2806:
2793:
2765:
2757:
2756:
2727:
2726:
2701:
2696:
2695:
2691:
2687:
2683:
2650:
2645:
2644:
2640:
2598:
2593:
2592:
2589:
2583:
2571:
2567:
2532:
2531:
2527:
2523:
2502:
2497:
2496:
2475:
2470:
2469:
2444:
2439:
2438:
2404:
2399:
2398:
2359:
2358:
2334:
2329:
2328:
2320:
2312:
2291:
2286:
2285:
2250:
2249:
2225:
2206:
2187:
2186:
2162:
2143:
2138:
2137:
2113:
2094:
2075:
2074:
2044:
2019:
1997:
1975:
1950:
1940:
1927:
1907:
1906:
1882:
1863:
1858:
1857:
1829:
1824:
1823:
1816:
1813:splitting field
1783:
1778:
1777:
1775:symmetric group
1770:
1759:
1756:rational number
1734:
1715:
1710:
1709:
1681:
1662:
1643:
1642:
1614:
1595:
1590:
1589:
1556:
1534:
1509:
1499:
1486:
1481:
1480:
1473:
1462:
1457:
1417:
1407:
1378:
1373:
1372:
1347:
1342:
1341:
1308:
1303:
1302:
1295:
1288:normal subgroup
1262:
1257:
1256:
1249:
1224:
1219:
1218:
1216:symmetric group
1208:
1205:
1179:
1174:
1173:
1171:symmetric group
1140:splitting field
1132:
1128:
1089:
1088:
1084:
1080:
1076:
1033:
1032:
998:
997:
993:
966:
965:
950:
899:
886:
881:
880:
859:
854:
853:
822:
817:
816:
795:
790:
789:
764:
759:
758:
731:
726:
725:
697:
692:
691:
664:
659:
658:
631:
616:
601:
600:
560:
559:
535:
516:
503:
498:
497:
476:
463:
458:
457:
433:
414:
401:
396:
395:
383:
360:
352:symmetric group
348:solvable groups
325:
309:Évariste Galois
288:
255:
250:
249:
239:
207:Newton's method
192:quartic formula
173:
165:quartic formula
105:
100:
99:
45:(also known as
35:
28:
23:
22:
15:
12:
11:
5:
4688:
4686:
4678:
4677:
4672:
4667:
4662:
4652:
4651:
4646:
4645:
4639:
4606:
4587:
4570:
4548:
4526:
4498:
4467:
4461:
4440:
4426:
4401:
4395:
4370:
4333:
4278:
4269:Ruffini, Paolo
4260:
4238:
4224:
4199:
4185:
4157:
4123:(6): 495–505,
4102:
4074:(1): 141–158,
4058:
4052:
4024:
3984:
3944:
3927:
3885:(3): 253–277,
3859:
3858:
3856:
3853:
3836:coefficients.
3833:Pierre Wantzel
3791:
3786:
3782:
3778:
3773:
3769:
3765:
3760:
3756:
3752:
3748:
3727:
3724:
3721:
3718:
3697:
3676:
3671:
3667:
3663:
3658:
3654:
3650:
3645:
3641:
3637:
3633:
3612:
3609:
3606:
3603:
3581:
3577:
3554:
3550:
3527:
3523:
3502:
3499:
3496:
3493:
3490:
3487:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3353:
3350:
3347:
3344:
3324:
3321:
3318:
3315:
3291:
3288:
3285:
3282:
3279:
3274:
3270:
3266:
3263:
3260:
3257:
3254:
3186:
3183:
3162:
3159:
3146:
3143:
3140:
3137:
3134:
3131:
3126:
3122:
3099:
3093:
3068:
3062:
3056:
3053:
3027:
3021:
2990:
2987:
2984:
2979:
2974:
2950:
2946:
2942:
2916:
2912:
2902:also contains
2886:
2862:
2859:
2856:
2851:
2846:
2824:
2821:
2818:
2813:
2809:
2805:
2800:
2796:
2792:
2789:
2786:
2783:
2780:
2777:
2772:
2768:
2764:
2742:
2738:
2734:
2712:
2706:
2671:
2668:
2665:
2662:
2657:
2653:
2625:
2622:
2619:
2616:
2613:
2610:
2605:
2601:
2582:
2579:
2554:
2551:
2548:
2545:
2542:
2539:
2509:
2505:
2482:
2478:
2451:
2447:
2420:
2415:
2409:
2386:
2383:
2379:
2375:
2372:
2369:
2366:
2346:
2341:
2337:
2298:
2294:
2269:
2266:
2263:
2260:
2257:
2237:
2232:
2228:
2224:
2221:
2218:
2213:
2209:
2205:
2201:
2197:
2194:
2174:
2169:
2165:
2161:
2158:
2155:
2150:
2146:
2125:
2120:
2116:
2112:
2109:
2106:
2101:
2097:
2093:
2089:
2085:
2082:
2071:
2070:
2059:
2056:
2051:
2047:
2043:
2040:
2037:
2034:
2031:
2026:
2022:
2018:
2015:
2012:
2009:
2004:
2000:
1996:
1993:
1988:
1985:
1982:
1978:
1974:
1971:
1968:
1963:
1960:
1957:
1953:
1947:
1943:
1939:
1934:
1930:
1926:
1923:
1920:
1917:
1914:
1889:
1885:
1881:
1878:
1875:
1870:
1866:
1845:
1840:
1834:
1794:
1788:
1741:
1737:
1733:
1730:
1727:
1722:
1718:
1693:
1688:
1684:
1680:
1677:
1674:
1669:
1665:
1661:
1657:
1653:
1650:
1636:indeterminates
1621:
1617:
1613:
1610:
1607:
1602:
1598:
1586:
1585:
1574:
1571:
1568:
1563:
1559:
1555:
1552:
1547:
1544:
1541:
1537:
1533:
1530:
1527:
1522:
1519:
1516:
1512:
1506:
1502:
1498:
1493:
1489:
1461:
1458:
1456:
1453:
1389:
1383:
1358:
1352:
1319:
1313:
1273:
1267:
1235:
1229:
1204:
1201:
1186:
1182:
1152:quotient group
1116:
1113:
1109:
1105:
1102:
1099:
1096:
1060:
1057:
1053:
1049:
1046:
1043:
1040:
1017:
1014:
1011:
1008:
1005:
981:
977:
973:
956:establishes a
949:
946:
914:
911:
906:
902:
898:
893:
889:
866:
862:
835:
832:
829:
825:
802:
798:
786:roots of unity
771:
767:
744:
741:
738:
734:
704:
700:
679:
676:
671:
667:
644:
641:
638:
634:
630:
623:
619:
613:
609:
588:
585:
582:
579:
576:
573:
570:
567:
547:
542:
538:
534:
529:
526:
523:
519:
515:
510:
506:
483:
479:
475:
470:
466:
454:
453:
440:
436:
432:
429:
426:
421:
417:
413:
408:
404:
359:
356:
324:
321:
276:
273:
270:
267:
262:
258:
231:multiplication
217:, that is, an
172:
169:
144:
143:
132:
129:
126:
123:
120:
117:
112:
108:
71:indeterminates
32:Abel's theorem
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4687:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4660:Galois theory
4658:
4657:
4655:
4642:
4636:
4632:
4628:
4624:
4620:
4616:
4610:
4607:
4600:
4599:
4591:
4588:
4585:
4581:
4574:
4571:
4567:
4563:
4559:
4555:
4551:
4549:1-4020-2186-0
4545:
4541:
4537:
4530:
4527:
4522:
4519:(in French),
4518:
4517:
4512:
4508:
4502:
4499:
4494:
4491:(in French),
4490:
4489:
4481:
4477:
4471:
4468:
4464:
4458:
4454:
4453:Basic Algebra
4450:
4444:
4441:
4437:
4433:
4429:
4423:
4419:
4415:
4408:
4406:
4402:
4398:
4392:
4388:
4384:
4383:Galois Theory
4380:
4374:
4371:
4366:
4362:
4358:
4354:
4353:Sylow, Ludwig
4347:
4343:
4337:
4334:
4330:
4326:
4322:
4318:
4314:
4310:
4306:
4302:
4298:
4294:
4293:
4285:
4283:
4279:
4274:
4270:
4264:
4261:
4256:
4252:
4248:
4242:
4239:
4235:
4231:
4227:
4225:0-262-66182-9
4221:
4217:
4214:, Cambridge:
4213:
4206:
4204:
4200:
4196:
4192:
4188:
4182:
4178:
4174:
4170:
4164:
4162:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4118:
4117:
4112:
4106:
4103:
4099:
4095:
4091:
4087:
4082:
4077:
4073:
4069:
4062:
4059:
4055:
4049:
4045:
4041:
4040:Galois Theory
4037:
4031:
4029:
4025:
4020:
4016:
4012:
4008:
4007:Sylow, Ludwig
4001:
3997:
3991:
3989:
3985:
3980:
3976:
3972:
3968:
3967:Sylow, Ludwig
3961:
3957:
3951:
3949:
3945:
3940:
3939:
3931:
3928:
3924:
3920:
3916:
3912:
3908:
3904:
3900:
3896:
3892:
3888:
3884:
3880:
3879:
3871:
3869:
3867:
3865:
3861:
3854:
3852:
3850:
3846:
3843:discovered a
3842:
3837:
3834:
3829:
3825:
3816:
3812:
3809:According to
3807:
3803:
3784:
3780:
3776:
3771:
3767:
3763:
3758:
3754:
3725:
3722:
3719:
3716:
3669:
3665:
3661:
3656:
3652:
3648:
3643:
3639:
3607:
3601:
3579:
3575:
3552:
3548:
3525:
3521:
3500:
3497:
3491:
3485:
3465:
3462:
3456:
3453:
3447:
3427:
3424:
3418:
3415:
3409:
3389:
3386:
3380:
3377:
3371:
3351:
3348:
3345:
3342:
3322:
3319:
3316:
3313:
3305:
3289:
3286:
3283:
3280:
3277:
3272:
3268:
3264:
3258:
3252:
3242:
3240:
3236:
3235:Paolo Ruffini
3228:
3224:
3223:Paolo Ruffini
3220:
3216:
3214:
3210:
3209:
3204:
3200:
3196:
3192:
3189:Around 1770,
3184:
3182:
3180:
3176:
3172:
3168:
3158:
3144:
3141:
3138:
3135:
3132:
3129:
3124:
3120:
3097:
3066:
3054:
3051:
3043:
3025:
3008:
2985:
2977:
2948:
2940:
2932:
2931:transposition
2929:, which is a
2914:
2910:
2884:
2857:
2849:
2819:
2816:
2811:
2807:
2803:
2798:
2794:
2784:
2781:
2778:
2775:
2770:
2766:
2740:
2732:
2710:
2669:
2666:
2663:
2660:
2655:
2651:
2637:
2623:
2620:
2617:
2614:
2611:
2608:
2603:
2599:
2591:The equation
2588:
2580:
2578:
2574:
2552:
2549:
2543:
2537:
2507:
2503:
2480:
2476:
2467:
2449:
2445:
2436:
2431:
2418:
2413:
2381:
2377:
2373:
2367:
2364:
2344:
2339:
2335:
2326:
2318:
2296:
2292:
2283:
2267:
2261:
2255:
2230:
2226:
2222:
2219:
2216:
2211:
2207:
2195:
2192:
2172:
2167:
2163:
2159:
2156:
2153:
2148:
2144:
2118:
2114:
2110:
2107:
2104:
2099:
2095:
2083:
2080:
2057:
2049:
2045:
2041:
2038:
2032:
2024:
2020:
2016:
2013:
2007:
2002:
1998:
1994:
1991:
1986:
1983:
1980:
1976:
1972:
1969:
1966:
1961:
1958:
1955:
1951:
1945:
1941:
1937:
1932:
1928:
1924:
1918:
1912:
1905:
1904:
1903:
1887:
1883:
1879:
1876:
1873:
1868:
1864:
1843:
1838:
1820:
1814:
1810:
1792:
1776:
1768:
1762:
1757:
1739:
1735:
1731:
1728:
1725:
1720:
1716:
1707:
1686:
1682:
1678:
1675:
1672:
1667:
1663:
1651:
1648:
1641:
1637:
1634:are distinct
1619:
1615:
1611:
1608:
1605:
1600:
1596:
1572:
1569:
1566:
1561:
1557:
1553:
1550:
1545:
1542:
1539:
1535:
1531:
1528:
1525:
1520:
1517:
1514:
1510:
1504:
1500:
1496:
1491:
1487:
1479:
1478:
1477:
1471:
1467:
1459:
1454:
1452:
1450:
1446:
1442:
1438:
1437:Galois theory
1434:
1430:
1426:
1420:
1414:
1410:
1405:
1387:
1356:
1339:
1335:
1317:
1298:
1293:
1289:
1271:
1255:
1252:has only the
1233:
1217:
1211:
1202:
1200:
1184:
1180:
1172:
1167:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1136:
1111:
1107:
1103:
1097:
1094:
1074:
1073:automorphisms
1055:
1051:
1047:
1041:
1038:
1031:
1015:
1012:
1009:
1006:
1003:
979:
975:
971:
963:
962:subextensions
959:
955:
947:
945:
943:
937:
935:
931:
925:
912:
904:
900:
891:
887:
864:
860:
851:
833:
830:
827:
823:
815:that extends
800:
796:
787:
769:
765:
742:
739:
736:
732:
723:
718:
702:
698:
677:
674:
669:
665:
642:
639:
636:
632:
628:
621:
617:
611:
607:
586:
583:
580:
577:
574:
571:
568:
565:
540:
536:
527:
524:
521:
517:
513:
508:
504:
481:
477:
473:
468:
464:
438:
434:
430:
427:
424:
419:
415:
411:
406:
402:
394:
393:
392:
389:
387:
380:
375:
373:
369:
365:
357:
355:
353:
349:
345:
341:
337:
332:
330:
329:Galois theory
322:
320:
318:
314:
313:Galois theory
310:
305:
302:
296:
294:
274:
271:
268:
265:
260:
256:
245:
243:
236:
232:
228:
224:
220:
216:
210:
208:
204:
200:
195:
193:
189:
188:cubic formula
185:
181:
177:
170:
168:
166:
162:
161:cubic formula
158:
153:
151:
150:
130:
127:
124:
121:
118:
115:
110:
106:
98:
97:
96:
94:
89:
85:
83:
79:
78:Paolo Ruffini
74:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
4618:
4609:
4597:
4590:
4573:
4535:
4529:
4520:
4514:
4501:
4492:
4486:
4470:
4452:
4443:
4413:
4382:
4379:Stewart, Ian
4373:
4360:
4336:
4296:
4290:
4272:
4263:
4254:
4241:
4211:
4172:
4120:
4114:
4105:
4071:
4067:
4061:
4039:
4036:Stewart, Ian
4014:
3974:
3937:
3930:
3882:
3876:
3838:
3823:
3808:
3804:
3513:, the roots
3243:
3232:
3226:
3206:
3195:permutations
3188:
3169:. This is a
3164:
2873:, the group
2638:
2590:
2572:
2432:
2282:permutations
2072:
1821:
1767:Galois group
1760:
1587:
1469:
1465:
1463:
1449:Galois group
1418:
1415:
1408:
1296:
1209:
1206:
1168:
1137:
1030:Galois group
960:between the
951:
938:
930:Galois group
926:
719:
455:
390:
376:
361:
344:Galois group
336:field theory
333:
326:
306:
297:
246:
211:
196:
174:
154:
147:
145:
87:
86:
75:
66:
63:coefficients
46:
42:
36:
4357:Lie, Sophus
4011:Lie, Sophus
3971:Lie, Sophus
3175:irreducible
3005:contains a
2755:factors as
1079:that leave
720:For having
496:such that
227:subtraction
59:degree five
53:to general
39:mathematics
4654:Categories
4566:1065.12001
4436:1333.12001
4234:1166.01010
4195:1333.12001
4153:0836.01015
3923:0471.01008
3855:References
2585:See also:
1439:is that a
1248:of degree
364:expression
219:expression
163:, and the
149:almost all
4495:: 417–433
4387:CRC Press
4344:(1881) ,
4329:121442989
4216:MIT Press
4081:1211.3588
4044:CRC Press
3998:(1881) ,
3958:(1881) ,
3915:123447349
3839:In 1963,
3720:±
3454:−
3416:−
3378:−
3346:−
3287:−
3278:−
3136:−
3130:−
2667:−
2661:−
2615:−
2609:−
2368:
2220:…
2157:…
2108:…
2042:−
2033:⋯
2017:−
1984:−
1970:⋯
1959:−
1877:…
1729:…
1676:…
1609:…
1543:−
1529:⋯
1518:−
1425:quadratic
1098:
1042:
1013:⊆
1007:⊆
831:−
740:−
640:−
629:∈
578:…
525:−
474:∈
431:⊆
428:⋯
425:⊆
412:⊆
266:−
184:antiquity
122:−
116:−
4617:(2014),
4509:(1845),
4478:(1846),
4359:(eds.),
4313:41133337
4271:(1799),
4013:(eds.),
3973:(eds.),
3899:41133596
3179:rational
2933:. Since
2725:. Since
1404:solvable
1402:are not
1154:that is
1144:solvable
932:that is
287:for any
235:division
223:addition
199:non-real
65:. Here,
4584:YouTube
4558:2110624
4523:: 57–65
4321:1554154
4253:(ed.),
4145:1336636
4137:2974763
4098:3230862
3907:0606270
3185:History
3044:. Thus
3007:5-cycle
2327:of the
2284:of the
1811:of the
1704:of the
1470:generic
1466:general
1433:quartic
1338:abelian
1294:). For
1160:abelian
1071:of the
1028:to the
171:Context
67:general
4637:
4564:
4556:
4546:
4459:
4434:
4424:
4393:
4327:
4319:
4311:
4232:
4222:
4193:
4183:
4151:
4143:
4135:
4096:
4050:
3921:
3913:
3905:
3897:
3567:, and
3478:, and
3229:, 1799
3213:thesis
3181:root.
2682:. Let
2575:> 4
1763:> 4
1588:where
1443:has a
1431:, and
1411:> 4
1334:simple
1299:> 4
1214:, the
1212:> 4
1156:cyclic
1148:normal
934:cyclic
388:root.
237:, and
159:, the
41:, the
4602:(PDF)
4483:(PDF)
4351:, in
4349:(PDF)
4325:S2CID
4309:JSTOR
4133:JSTOR
4076:arXiv
4005:, in
4003:(PDF)
3965:, in
3963:(PDF)
3911:S2CID
3895:JSTOR
2898:then
2323:is a
1769:over
1754:with
1640:field
1429:cubic
1290:(see
996:such
848:by a
599:with
379:field
323:Proof
4635:ISBN
4544:ISBN
4457:ISBN
4422:ISBN
4391:ISBN
4220:ISBN
4181:ISBN
4048:ISBN
3498:>
3463:<
3425:>
3387:<
2639:Let
2464:are
2433:The
2280:The
2185:and
2073:Let
1464:The
1406:for
1371:and
1207:For
952:The
784:-th
675:>
558:for
4627:doi
4582:on
4562:Zbl
4432:Zbl
4301:doi
4230:Zbl
4191:Zbl
4149:Zbl
4125:doi
4121:102
4086:doi
3919:Zbl
3887:doi
3594:of
2945:mod
2835:in
2737:mod
2643:be
2526:to
2365:Gal
1708:in
1468:or
1421:≤ 4
1332:is
1095:Gal
1087:of
1075:of
1039:Gal
879:as
57:of
37:In
4656::
4633:,
4625:,
4560:,
4554:MR
4552:,
4542:,
4538:,
4513:,
4493:XI
4485:,
4430:,
4420:,
4404:^
4389:,
4355:;
4323:,
4317:MR
4315:,
4307:,
4295:,
4281:^
4228:,
4218:,
4202:^
4189:,
4179:,
4160:^
4147:,
4141:MR
4139:,
4131:,
4119:,
4094:MR
4092:,
4084:,
4072:17
4070:,
4046:,
4027:^
4009:;
3987:^
3969:;
3947:^
3917:,
3909:,
3903:MR
3901:,
3893:,
3883:22
3881:,
3863:^
3851:.
3717:10
3540:,
3440:,
3402:,
3343:10
3314:10
3290:20
3281:15
3225:,
2577:.
2315:.
1427:,
1135:.
944:.
936:.
678:1.
386:th
244:.
233:,
229:,
225:,
73:.
4629::
4521:4
4303::
4297:8
4127::
4088::
4078::
3889::
3820:4
3790:)
3785:3
3781:r
3777:,
3772:2
3768:r
3764:,
3759:1
3755:r
3751:(
3747:Q
3726:i
3723:5
3696:R
3675:)
3670:3
3666:r
3662:,
3657:2
3653:r
3649:,
3644:1
3640:r
3636:(
3632:Q
3611:)
3608:x
3605:(
3602:P
3580:3
3576:r
3553:2
3549:r
3526:1
3522:r
3501:0
3495:)
3492:5
3489:(
3486:P
3466:0
3460:)
3457:1
3451:(
3448:P
3428:0
3422:)
3419:2
3413:(
3410:P
3390:0
3384:)
3381:3
3375:(
3372:P
3352:i
3349:5
3323:i
3320:5
3317:+
3284:x
3273:3
3269:x
3265:=
3262:)
3259:x
3256:(
3253:P
3145:0
3142:=
3139:1
3133:x
3125:5
3121:x
3098:5
3092:S
3067:5
3061:S
3055:=
3052:G
3026:5
3020:S
3003:G
2989:]
2986:x
2983:[
2978:3
2973:F
2949:3
2941:q
2915:3
2911:g
2900:G
2885:g
2875:G
2861:]
2858:x
2855:[
2850:2
2845:F
2823:)
2820:1
2817:+
2812:2
2808:x
2804:+
2799:3
2795:x
2791:(
2788:)
2785:1
2782:+
2779:x
2776:+
2771:2
2767:x
2763:(
2741:2
2733:q
2711:5
2705:S
2692:G
2688:q
2684:G
2670:1
2664:x
2656:5
2652:x
2641:q
2624:0
2621:=
2618:1
2612:x
2604:5
2600:x
2573:n
2568:n
2553:0
2550:=
2547:)
2544:x
2541:(
2538:P
2528:K
2524:F
2508:i
2504:b
2481:i
2477:a
2450:i
2446:b
2419:.
2414:n
2408:S
2385:)
2382:K
2378:/
2374:H
2371:(
2345:,
2340:i
2336:x
2321:K
2313:H
2297:i
2293:x
2268:.
2265:)
2262:x
2259:(
2256:P
2236:)
2231:n
2227:b
2223:,
2217:,
2212:1
2208:b
2204:(
2200:Q
2196:=
2193:K
2173:,
2168:n
2164:x
2160:,
2154:,
2149:1
2145:x
2124:)
2119:n
2115:x
2111:,
2105:,
2100:1
2096:x
2092:(
2088:Q
2084:=
2081:H
2058:.
2055:)
2050:n
2046:x
2039:x
2036:(
2030:)
2025:1
2021:x
2014:x
2011:(
2008:=
2003:n
1999:b
1995:+
1992:x
1987:1
1981:n
1977:b
1973:+
1967:+
1962:1
1956:n
1952:x
1946:1
1942:b
1938:+
1933:n
1929:x
1925:=
1922:)
1919:x
1916:(
1913:P
1888:n
1884:x
1880:,
1874:,
1869:1
1865:x
1844:,
1839:n
1833:S
1817:F
1793:n
1787:S
1771:F
1761:n
1740:n
1736:a
1732:,
1726:,
1721:1
1717:a
1692:)
1687:n
1683:a
1679:,
1673:,
1668:1
1664:a
1660:(
1656:Q
1652:=
1649:F
1620:n
1616:a
1612:,
1606:,
1601:1
1597:a
1573:,
1570:0
1567:=
1562:n
1558:a
1554:+
1551:x
1546:1
1540:n
1536:a
1532:+
1526:+
1521:1
1515:n
1511:x
1505:1
1501:a
1497:+
1492:n
1488:x
1474:n
1419:n
1409:n
1388:n
1382:S
1357:n
1351:A
1318:n
1312:A
1297:n
1272:n
1266:A
1250:n
1234:n
1228:S
1210:n
1185:5
1181:S
1133:H
1129:F
1115:)
1112:E
1108:/
1104:F
1101:(
1085:H
1081:K
1077:F
1059:)
1056:K
1052:/
1048:F
1045:(
1016:F
1010:K
1004:E
994:K
980:E
976:/
972:F
913:.
910:)
905:i
901:x
897:(
892:i
888:K
865:i
861:F
834:1
828:i
824:F
801:i
797:K
770:i
766:n
743:1
737:i
733:F
703:k
699:F
670:i
666:n
643:1
637:i
633:F
622:i
618:n
612:i
608:x
587:,
584:k
581:,
575:,
572:1
569:=
566:i
546:)
541:i
537:x
533:(
528:1
522:i
518:F
514:=
509:i
505:F
482:i
478:F
469:i
465:x
439:k
435:F
420:1
416:F
407:0
403:F
384:n
289:n
275:0
272:=
269:1
261:n
257:x
240:n
131:0
128:=
125:1
119:x
111:5
107:x
34:.
20:)
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