2191:
31:
1219:
1644:
8781:
1633:
810:
2186:{\displaystyle {\begin{aligned}\Delta ={}&-128p^{2}r^{4}+3125s^{4}-72p^{4}qrs+560p^{2}qr^{2}s+16p^{4}r^{3}+256r^{5}+108p^{5}s^{2}\\&-1600qr^{3}s+144pq^{2}r^{3}-900p^{3}rs^{2}+2000pr^{2}s^{2}-3750pqs^{3}+825p^{2}q^{2}s^{2}\\&+2250q^{2}rs^{2}+108q^{5}s-27q^{4}r^{2}-630pq^{3}rs+16p^{3}q^{3}s-4p^{3}q^{2}r^{2}.\end{aligned}}}
8420:
1253:
3261:
It follows that one may need four different square roots for writing all the roots of a solvable quintic. Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the
1214:{\displaystyle {\begin{aligned}p&={\frac {5ac-2b^{2}}{5a^{2}}}\\q&={\frac {25a^{2}d-15abc+4b^{3}}{25a^{3}}}\\r&={\frac {125a^{3}e-50a^{2}bd+15ab^{2}c-3b^{4}}{125a^{4}}}\\s&={\frac {3125a^{4}f-625a^{3}be+125a^{2}b^{2}d-25ab^{3}c+4b^{5}}{3125a^{5}}}\end{aligned}}}
2196:
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 resolvent.
3761:
3420:
2424:
548:
As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only irreducible quintic equations are considered in the remainder of this section, and the term "quintic" will refer only to irreducible quintics. A
5094:
424:
in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions. However, there is no
2988:
8123:
8776:{\displaystyle {\begin{aligned}&a=\pm (M_{S}+M_{E}),\\&b=+(M_{S}+M_{E})3R,\\&c=\pm (M_{S}+M_{E})3R^{2},\\&d=+(M_{E}\mp M_{E})R^{3}\ (thus\ d=0\ for\ L_{2}),\\&e=\pm M_{E}2R^{4},\\&f=\mp M_{E}R^{5}\end{aligned}}}
4016:
2600:
1628:{\displaystyle {\begin{aligned}P={}&z^{3}-z^{2}(20r+3p^{2})-z(8p^{2}r-16pq^{2}-240r^{2}+400sq-3p^{4})\\&-p^{6}+28p^{4}r-16p^{3}q^{2}-176p^{2}r^{2}-80p^{2}sq+224prq^{2}-64q^{4}\\&+4000ps^{2}+320r^{3}-1600rsq\end{aligned}}}
3206:
4407:
8307:
7632:
for the quintic, which is discussed in Dummit. Indeed, if an irreducible quintic has all roots real, no root can be expressed purely in terms of real radicals (as is true for all polynomial degrees that are not powers of 2).
7143:
3879:
2698:
7889:
5293:
3123:
7284:
6965:
6792:
5431:
6597:
6432:
8930:
5492:
166:
6270:
5614:
5553:
368:
7491:
4569:
8425:
5702:
1649:
1258:
815:
678:
8409:
6687:
6027:
5939:
6883:
5860:
6522:
4933:
433:, first asserted in 1799 and completely proven in 1824. This result also holds for equations of higher degree. An example of a quintic whose roots cannot be expressed in terms of radicals is
6357:
3526:
6214:
5781:
4639:
5173:
4987:
4230:
4849:
3291:
3256:
2259:
798:
7608:
7398:
4779:
4709:
447:
Some quintics may be solved in terms of radicals. However, the solution is generally too complicated to be used in practice. Instead, numerical approximations are calculated using a
6090:
5354:
7714:, and the elliptic modular functions that are featured in Hermite's solution, giving an explanation for why they should appear at all, and developed his own solution in terms of
4298:
4997:
4489:
2833:
536:
7178:
6712:
5198:
7999:
2819:
is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers
3902:
2450:
4135:-roots in term of one of them. Galois theory shows that this is always theoretically possible, even if the resulting formula may be too large to be of any use.
3134:
1224:
Both quintics are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial
9340:
4309:
8202:
3050:. The other solutions may then be obtained either by changing the fifth root or by multiplying all the occurrences of the fifth root by the same power of a
7967:
are the solutions to the following equations, where the gravitational forces of two masses on a third (for example, Sun and Earth on satellites such as
4144:
are rational (as in the first example of this section) or some are zero. In these cases, the formula for the roots is much simpler, as for the solvable
6981:
3772:
2732:. Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second.
3128:
In fact, all four primitive fifth roots of unity may be obtained by changing the signs of the square roots appropriately; namely, the expression
2611:
9378:
7762:
5203:
3060:
9348:
7185:
9412:
7626:, there are solvable quintics which have five real roots all of whose solutions in radicals involve roots of complex numbers. This is
6893:
6720:
5364:
429:(that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the
9300:
6531:
6366:
459:
Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is
8873:
5441:
59:
9453:
9179:
Kronecker, Leopold (1858). "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite".
6223:
5563:
5502:
389:
were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the
448:
8822:
273:
7408:
4501:
7715:
5624:
581:
8315:
6605:
5949:
5870:
4439:
There are infinitely many solvable quintics in Bring–Jerrard form which have been parameterized in a preceding section.
3756:{\displaystyle -cx={\sqrt{(a+c)^{2}(b-c)}}+{\sqrt{(-a+c)(b-c)^{2}}}+{\sqrt{(a+c)(b+c)^{2}}}-{\sqrt{(-a+c)^{2}(b+c)}}\,,}
9633:
7983:
6801:
5791:
6440:
4864:
9358:
9296:
9278:
9233:
Leopold
Kronecker, "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite",
9224:
6278:
6132:
5712:
4575:
3415:{\displaystyle x=1+{\sqrt{2}}-\left({\sqrt{2}}\right)^{2}+\left({\sqrt{2}}\right)^{3}-\left({\sqrt{2}}\right)^{4}.}
2419:{\displaystyle x^{5}+{\frac {5\mu ^{4}(4\nu +3)}{\nu ^{2}+1}}x+{\frac {4\mu ^{5}(2\nu +1)(4\nu +3)}{\nu ^{2}+1}}=0}
5104:
4943:
4154:
572:
found a general criterion for determining whether any given quintic is solvable. This criterion is the following.
4785:
3425:
An example of a more complicated (although small enough to be written here) solution is the unique real root of
3214:
723:
7501:
7297:
4715:
7993:) provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun:
7731:
4645:
9694:
7895:
7950:
of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic.
6037:
5301:
9628:
9612:
35:
9160:
Brioschi, Francesco (1858). "Sul Metodo di
Kronecker per la Risoluzione delle Equazioni di Quinto Grado".
3018:
of all permutations of a five element set, which is solvable if and only if it is a subgroup of the group
9648:
9331:
Ehrenfried Walter von
Tschirnhaus, "A method for removing all intermediate terms from a given equation,"
9324:
Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of
Tschirnhaus, Bring and Jerrard,"
4244:
5089:{\displaystyle x^{5}-{\frac {22}{5}}x^{3}-{\frac {11}{25}}x^{2}+{\frac {462}{125}}x+{\frac {979}{3125}}}
9493:
7972:
7749:
684:
2983:{\displaystyle a={\frac {5l(3l^{5}-4m)}{m^{2}+l^{10}}}\qquad b={\frac {4(11l^{5}+2m)}{m^{2}+l^{10}}}.}
9446:
9418:
7679:
4445:
2605:
The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression
2204:
described how to solve a solvable quintic equation, without providing an explicit formula; in 2004,
9638:
430:
390:
9577:
4424:
496:
17:
9364:
8118:{\displaystyle {\frac {GmM_{S}}{(R\pm r)^{2}}}\pm {\frac {GmM_{E}}{r^{2}}}=m\omega ^{2}(R\pm r)}
9689:
9607:
9602:
9597:
9475:
9281:. Discusses Galois Theory in general including a proof of insolvability of the general quintic.
8147:
7687:
7151:
6693:
5179:
2243:
2232:
460:
220:
50:
4038:
with rational coefficients is solvable in radicals, then one can define an auxiliary equation
9587:
9567:
9128:
4081:
717:, which depresses the quintic (that is, removes the term of degree four), gives the equation
556:
To characterize solvable quintics, and more generally solvable polynomials of higher degree,
553:
is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals.
4011:{\displaystyle y^{4}+4y^{3}+{\frac {4}{5}}y^{2}-{\frac {8}{5^{3}}}y-{\frac {1}{5^{5}}}=0\,.}
9653:
9572:
9439:
4235:
where the auxiliary equation has two zero roots and reduces, by factoring them out, to the
3043:
2595:{\displaystyle x^{5}+{\frac {5e^{4}(4c+3)}{c^{2}+1}}x+{\frac {-4e^{5}(2c-11)}{c^{2}+1}}=0.}
2246:
quintic with rational coefficients in Bring–Jerrard form is solvable if and only if either
426:
6975:
There are also two parameterized families of solvable quintics: The Kondo–Brumer quintic,
2238:
During the second half of the 19th century, John Stuart
Glashan, George Paxton Young, and
8:
9466:
8972:
7628:
4442:
Up to the scaling of the variable, there are exactly five solvable quintics of the shape
3201:{\displaystyle {\frac {\alpha {\sqrt {-10-2\beta {\sqrt {5}}}}+\beta {\sqrt {5}}-1}{4}},}
2201:
196:
9406:
9307: (archived 31 March 2010)) gives a description of the solution of solvable quintics
9284:
9108:
9684:
9643:
9510:
9505:
9431:
9372:
9094:
9057:
9017:
A. Cayley, "On a new auxiliary equation in the theory of equation of the fifth order",
8994:"Equations of the Bring–Jerrard Form, the Golden Section, and Square Fibonacci Numbers"
7924:
7699:
4402:{\displaystyle x_{k}=\omega ^{k}{\sqrt{y_{i}}}-{\frac {a}{\omega ^{k}{\sqrt{y_{i}}}}},}
4236:
4091:
413:
402:
9086:
8302:{\displaystyle \omega ^{2}={\frac {4\pi ^{2}}{P^{2}}}={\frac {G(M_{S}+M_{E})}{R^{3}}}}
6101:
An infinite sequence of solvable quintics may be constructed, whose roots are sums of
557:
9488:
9400:
9354:
9292:
9274:
9220:
7928:
7920:
7691:
2786:
in the
Spearman-Williams parameterization allows one to not exclude the special case
231:
9534:
9529:
9049:
George Paxton Young, "Solvable
Quintic Equations with Commensurable Coefficients",
9025:
8155:
7947:
7753:
3894:
421:
386:
239:
226:
Because they have an odd degree, normal quintic functions appear similar to normal
9663:
9551:
9546:
9498:
9483:
9304:
8967:
8962:
8167:
7916:
7723:
7671:
7138:{\displaystyle x^{5}+(a-3)\,x^{4}+(-a+b+3)\,x^{3}+(a^{2}-a-1-2b)\,x^{2}+b\,x+a=0}
4428:
3007:
409:
200:
3874:{\displaystyle x={\sqrt{y_{1}}}+{\sqrt{y_{2}}}+{\sqrt{y_{3}}}+{\sqrt{y_{4}}}\,,}
545:
is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible.
9522:
9517:
9244:
Blair
Spearman and Kenneth S. Williams, "Characterization of solvable quintics
9214:
9141:
Hermite, Charles (1858). "Sur la résolution de l'équation du cinquième degré".
8993:
7683:
7675:
7642:
7623:
6106:
3047:
3003:
417:
382:
227:
208:
7915:
The roots of this equation cannot be expressed by radicals. However, in 1858,
2216:
There are several parametric representations of solvable quintics of the form
9678:
9216:
Lectures on the
Icosahedron and the Solution of Equations of the Fifth Degree
7935:
7743:
7711:
3051:
3006:. In the case of irreducible quintics, the Galois group is a subgroup of the
2205:
569:
565:
7674:
showed that the Bring radical could be characterized in terms of the Jacobi
9658:
9029:
7695:
7646:
3510:
2999:
2693:{\displaystyle b={\frac {4}{5}}\left(a+20\pm 2{\sqrt {(20-a)(5+a)}}\right)}
561:
9040:
This formulation of Cayley's result is extracted from Lazard (2004) paper.
3042:
If the quintic is solvable, one of the solutions may be represented by an
9425:
9344:
8864:
7707:
7703:
405:(zeros) of a given polynomial has been a prominent mathematical problem.
204:
42:
9462:
9061:
7968:
7884:{\displaystyle x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\,}
5288:{\displaystyle {\sqrt{2}}-{\sqrt{2}}^{2}+{\sqrt{2}}^{3}-{\sqrt{2}}^{4}}
2239:
541:
has solutions in radicals if and only if it has an integer solution or
235:
216:
9201:
Charles
Hermite, "Sur la résolution de l'équation du cinquème degré",
3118:{\displaystyle {\frac {{\sqrt {-10-2{\sqrt {5}}}}+{\sqrt {5}}-1}{4}}.}
9592:
7682:, using an approach similar to the more familiar approach of solving
4145:
3262:
form of the first solution may be rather simple, as for the equation
2444:
In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,
381:
th roots) was a major problem in algebra from the 16th century, when
30:
9127:, Section 1.6, Additional Topic: Klein's Theory of the Icosahedron,
7279:{\displaystyle x^{5}-5\,p\left(2\,x^{3}+a\,x^{2}+b\,x\right)-p\,c=0}
4854:
Paxton Young (1888) gave a number of examples of solvable quintics:
9582:
374:
34:
Graph of a polynomial of degree 5, with 3 real zeros (roots) and 4
9413:
A method for removing all intermediate terms from a given equation
9409:– a method for solving solvable quintics due to David S. Dummit.
7919:
published the first known solution of this equation in terms of
4117:-th roots may not be computed independently (this would provide
9388:
Finite Möbius groups, minimal immersions of spheres, and moduli
7698:, developed a simpler way of deriving Hermite's result, as had
4303:
such that the five roots of the de Moivre quintic are given by
3046:
involving a fifth root and at most two square roots, generally
215:
is nonzero. In other words, a quintic function is defined by a
6960:{\displaystyle \sum _{k=0}^{13}e^{\frac {2i\pi (23)^{k}}{71}}}
6787:{\displaystyle \sum _{k=0}^{11}e^{\frac {2i\pi (21)^{k}}{61}}}
9162:
Atti Dell'i. R. Istituto Lombardo di Scienze, Lettere ed Arti
8189:
5426:{\displaystyle x^{5}-5x^{3}+{\frac {85}{8}}x-{\frac {13}{2}}}
3258:, yields the four distinct primitive fifth roots of unity.
9415:- a recent English translation of Tschirnhaus' 1683 paper.
9339:
Lazard, Daniel (2004). "Solving quintics in radicals". In
6592:{\displaystyle \sum _{k=0}^{7}e^{\frac {2i\pi 3^{k}}{41}}}
6427:{\displaystyle \sum _{k=0}^{5}e^{\frac {2i\pi 6^{k}}{31}}}
4427:. This can be easily generalized to construct a solvable
4056:, also with rational coefficients, such that each root of
9219:. Translated by Morrice, George Gavin. Trübner & Co.
9019:
Philosophical Transactions of the Royal Society of London
8855:), then the quintic equation can be greatly reduced and L
8193:
7706:
came up with a method that relates the symmetries of the
9461:
8925:{\displaystyle r\approx R{\sqrt{\frac {M_{E}}{3M_{S}}}}}
7649:(also known as Bring radicals), the unique real root of
5487:{\displaystyle x^{5}+{\frac {20}{17}}x+{\frac {21}{17}}}
161:{\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,}
6265:{\displaystyle 2\cos \left({\frac {2k\pi }{11}}\right)}
5609:{\displaystyle x^{5}+{\frac {10}{13}}x+{\frac {3}{13}}}
5548:{\displaystyle x^{5}-{\frac {4}{13}}x+{\frac {29}{65}}}
8848:) is much smaller than the mass of the larger object (
7938:
for details on these solutions and some related ones.
7512:
7308:
4129:). Thus a correct solution needs to express all these
396:
9289:
Galois theory for beginners: A historical perspective
8876:
8423:
8318:
8205:
8002:
7941:
7765:
7504:
7411:
7300:
7188:
7154:
6984:
6896:
6804:
6723:
6696:
6608:
6534:
6443:
6369:
6281:
6226:
6135:
6040:
5952:
5873:
5794:
5715:
5627:
5566:
5505:
5444:
5367:
5304:
5206:
5182:
5107:
5000:
4946:
4867:
4788:
4718:
4648:
4578:
4504:
4448:
4312:
4247:
4157:
3905:
3775:
3529:
3294:
3217:
3137:
3063:
2998:
A polynomial equation is solvable by radicals if its
2836:
2614:
2453:
2262:
1647:
1256:
813:
726:
584:
499:
363:{\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,}
276:
230:
when graphed, except they may possess one additional
62:
7756:, reduces the general quintic equation of the form
7486:{\displaystyle b=\ell \,(4m^{2}+a^{2})-5p-2m^{2}\;,}
4564:{\displaystyle x^{5}-2s^{3}x^{2}-{\frac {s^{5}}{5}}}
4419:
is any root of the auxiliary quadratic equation and
5697:{\displaystyle x^{5}+110(5x^{3}+60x^{2}+800x+8320)}
673:{\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0,}
8924:
8775:
8404:{\displaystyle ar^{5}+br^{4}+cr^{3}+dr^{2}+er+f=0}
8403:
8301:
8117:
7883:
7645:demonstrated that quintics can be solved by using
7602:
7485:
7392:
7278:
7172:
7137:
6959:
6877:
6786:
6706:
6682:{\displaystyle x^{5}+x^{4}-24x^{3}-17x^{2}+41x-13}
6681:
6591:
6516:
6426:
6351:
6264:
6208:
6084:
6022:{\displaystyle x^{5}+110(5x^{3}+20x^{2}-360x+800)}
6021:
5934:{\displaystyle x^{5}-50x^{3}-600x^{2}-2000x-11200}
5933:
5854:
5775:
5696:
5608:
5547:
5486:
5425:
5348:
5287:
5192:
5167:
5088:
4981:
4927:
4843:
4773:
4703:
4633:
4563:
4483:
4401:
4292:
4224:
4010:
3873:
3755:
3414:
3250:
3200:
3117:
2982:
2692:
2594:
2418:
2185:
1627:
1213:
792:
672:
530:
362:
160:
9421:The Quintic, the Icosahedron, and Elliptic Curves
9328:, Vol. 37, No. 3, September 2003, pp. 90–94.
6878:{\displaystyle x^{5}+x^{4}-28x^{3}+37x^{2}+25x+1}
5855:{\displaystyle x^{5}-40x^{3}+160x^{2}+1000x-5888}
2211:
9676:
8838:are usually given as 1.5 million km from Earth.
6517:{\displaystyle x^{5}+x^{4}-16x^{3}+5x^{2}+21x-9}
4928:{\displaystyle x^{5}-10x^{3}-20x^{2}-1505x-7412}
9403:– more details on methods for solving Quintics.
6352:{\displaystyle x^{5}+x^{4}-12x^{3}-21x^{2}+x+5}
9484:Zero polynomial (degree undefined or −1 or −∞)
7737:
7732:Icosahedral symmetry § Related geometries
6209:{\displaystyle x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1}
5776:{\displaystyle x^{5}-20x^{3}-80x^{2}-150x-656}
4431:and other odd degrees, not necessarily prime.
2993:
9447:
9301:The solution of equations of the fifth degree
8991:
8309:and rearranging all terms yields the quintic
4634:{\displaystyle x^{5}-100s^{3}x^{2}-1000s^{5}}
5168:{\displaystyle x^{5}+20x^{3}+20x^{2}+30x+10}
4982:{\displaystyle x^{5}+{\frac {625}{4}}x+3750}
4225:{\displaystyle x^{5}+5ax^{3}+5a^{2}x+b=0\,,}
3245:
3230:
9335:, Vol. 37, No. 1, March 2003, pp. 1–3.
7148:and the family depending on the parameters
4844:{\displaystyle x^{5}-25s^{3}x^{2}-300s^{5}}
9454:
9440:
9377:: CS1 maint: location missing publisher (
7596:
7479:
7386:
4434:
3251:{\displaystyle \alpha ,\beta \in \{-1,1\}}
793:{\displaystyle y^{5}+py^{3}+qy^{2}+ry+s=0}
9235:Comptes Rendus de l'Académie des Sciences
9181:Comptes Rendus de l'Académie des Sciences
9178:
9143:Comptes Rendus de l'Académie des Sciences
7880:
7603:{\displaystyle c={\tfrac {1}{2}}\left\;.}
7590:
7528:
7421:
7393:{\displaystyle p={\tfrac {1}{4}}\left\;,}
7380:
7324:
7266:
7251:
7234:
7217:
7205:
7119:
7102:
7051:
7013:
4774:{\displaystyle x^{5}-5s^{3}x^{2}+15s^{5}}
4286:
4218:
4138:It is possible that some of the roots of
4004:
3867:
3749:
359:
157:
9159:
8188:are the respective masses of satellite,
8166:the distance Sun to Earth (that is, the
8162:the distance of the satellite to Earth,
4704:{\displaystyle x^{5}-5s^{3}x^{2}-3s^{5}}
560:developed techniques which gave rise to
29:
9291:, American Mathematical Society, 2006.
9140:
8863:are at approximately the radius of the
7730:, as studied by Klein and discussed in
3031:, generated by the cyclic permutations
14:
9677:
9338:
9134:
9072:
7615:
6085:{\displaystyle x^{5}-20x^{3}+170x+208}
5349:{\displaystyle x^{5}-20x^{3}+250x-400}
3285:, for which the only real solution is
490:. For example, it has been shown that
449:root-finding algorithm for polynomials
373:Solving quintic equations in terms of
234:and one additional local minimum. The
9435:
9273:2nd Edition, Chapman and Hall, 1989.
9212:
9120:
7752:, which may be computed by solving a
9385:
9124:
9123:); a modern exposition is given in (
7718:. Similar phenomena occur in degree
7716:generalized hypergeometric functions
454:
8841:If the mass of the smaller object (
4293:{\displaystyle y^{2}+by-a^{5}=0\,,}
4100:and its roots can be used to solve
2804:are rational numbers, the equation
397:Finding roots of a quintic equation
24:
9084:
8787:Solving these two quintics yields
7942:Application to celestial mechanics
7636:
1652:
25:
9706:
9394:
7953:More precisely, the locations of
7946:Solving for the locations of the
2242:gave such a parameterization: an
8992:Elia, M.; Filipponi, P. (1998).
7931:came upon equivalent solutions.
2208:wrote out a three-page formula.
9350:The Legacy of Niels Henrik Abel
9172:
9153:
9091:with interesting Galois groups"
9051:American Journal of Mathematics
6700:
5186:
4021:More generally, if an equation
2908:
2793:, giving the following result:
27:Polynomial function of degree 5
9209::5–21, Gauthier-Villars, 1908.
9113:
9101:
9078:
9066:
9043:
9034:
9011:
8985:
8689:
8637:
8621:
8595:
8562:
8536:
8510:
8484:
8464:
8438:
8283:
8257:
8112:
8100:
8037:
8024:
7571:
7556:
7547:
7532:
7451:
7422:
7364:
7335:
7099:
7065:
7048:
7027:
7010:
6998:
6941:
6934:
6768:
6761:
6016:
5969:
5691:
5644:
4484:{\displaystyle x^{5}+ax^{2}+b}
3738:
3726:
3717:
3701:
3678:
3665:
3662:
3650:
3627:
3614:
3611:
3596:
3579:
3567:
3558:
3545:
3513:. Then the only real solution
2946:
2921:
2877:
2852:
2680:
2668:
2665:
2653:
2562:
2547:
2498:
2483:
2386:
2371:
2368:
2353:
2307:
2292:
2212:Quintics in Bring–Jerrard form
1410:
1328:
1319:
1294:
72:
66:
13:
1:
9644:Horner's method of evaluation
9260:American Mathematical Monthly
9195:
4080:-th roots were introduced by
568:. Applying these techniques,
9401:Mathworld - Quintic Equation
9353:. Berlin. pp. 207–225.
7690:. At around the same time,
4425:primitive 5th roots of unity
7:
9649:Polynomial identity testing
8956:
8823:Sun–Earth Lagrangian points
7738:Solving with Bring radicals
3052:primitive 5th root of unity
2994:Roots of a solvable quintic
531:{\displaystyle x^{5}-x-r=0}
238:of a quintic function is a
10:
9711:
8128:The ± sign corresponds to
7973:James Webb Space Telescope
7923:. At around the same time
7750:Tschirnhaus transformation
7741:
7680:elliptic modular functions
4068:-th roots of the roots of
3893:are the four roots of the
685:Tschirnhaus transformation
9621:
9560:
9473:
9407:Solving Solvable Quintics
9203:Œuvres de Charles Hermite
9109:Solving Solvable Quintics
8953:in the Sun-Earth system.
8199:Using Kepler's Third Law
7896:Bring–Jerrard normal form
7173:{\displaystyle a,\ell ,m}
6707:{\displaystyle ~\qquad ~}
5193:{\displaystyle ~\qquad ~}
1243:, has a rational root in
8978:
4084:, and their products by
9634:Greatest common divisor
9001:The Fibonacci Quarterly
8170:of Earth's orbit), and
7688:trigonometric functions
4435:Other solvable quintics
9506:Quadratic function (2)
9030:10.1098/rstl.1861.0014
8926:
8777:
8405:
8303:
8148:gravitational constant
8119:
7885:
7604:
7487:
7394:
7280:
7174:
7139:
6961:
6917:
6879:
6788:
6744:
6708:
6683:
6593:
6555:
6518:
6428:
6390:
6353:
6266:
6210:
6122:being a prime number:
6086:
6023:
5935:
5856:
5777:
5698:
5610:
5549:
5488:
5427:
5350:
5289:
5194:
5169:
5090:
4983:
4929:
4845:
4775:
4705:
4635:
4565:
4495:is a scaling factor):
4485:
4403:
4294:
4226:
4012:
3875:
3757:
3416:
3252:
3202:
3119:
2984:
2694:
2596:
2420:
2187:
1629:
1215:
794:
674:
532:
364:
162:
38:
9489:Constant function (0)
9213:Klein, Felix (1888).
8927:
8778:
8406:
8304:
8120:
7886:
7678:and their associated
7605:
7488:
7395:
7281:
7175:
7140:
6962:
6897:
6880:
6789:
6724:
6709:
6684:
6594:
6535:
6519:
6429:
6370:
6354:
6267:
6211:
6087:
6024:
5936:
5857:
5778:
5699:
5611:
5550:
5489:
5428:
5351:
5290:
5195:
5170:
5091:
4984:
4930:
4846:
4776:
4706:
4636:
4566:
4486:
4404:
4295:
4227:
4094:. The computation of
4082:Joseph-Louis Lagrange
4013:
3876:
3766:or, equivalently, by
3758:
3417:
3253:
3203:
3120:
2985:
2695:
2597:
2421:
2253:or it may be written
2188:
1630:
1216:
795:
675:
533:
365:
163:
33:
9695:Polynomial functions
9622:Tools and algorithms
9542:Quintic function (5)
9530:Quartic function (4)
9467:polynomial functions
9386:Tóth, Gábor (2002),
8874:
8421:
8316:
8203:
8000:
7763:
7502:
7409:
7298:
7186:
7152:
6982:
6894:
6802:
6721:
6694:
6606:
6532:
6441:
6367:
6279:
6224:
6133:
6038:
5950:
5871:
5792:
5713:
5625:
5564:
5503:
5442:
5365:
5302:
5204:
5180:
5105:
4998:
4944:
4865:
4786:
4716:
4646:
4576:
4502:
4446:
4310:
4245:
4155:
4090:are commonly called
3903:
3773:
3527:
3292:
3215:
3135:
3061:
3044:algebraic expression
2834:
2612:
2451:
2260:
1645:
1254:
811:
724:
582:
497:
431:Abel–Ruffini theorem
427:algebraic expression
391:Abel–Ruffini theorem
274:
60:
9552:Septic equation (7)
9547:Sextic equation (6)
9494:Linear function (1)
9367:on January 6, 2005.
9341:Olav Arnfinn Laudal
9333:ACM SIGSAM Bulletin
9326:ACM SIGSAM Bulletin
8973:Theory of equations
8945:for satellites at L
7629:casus irreducibilis
7617:Casus irreducibilis
4491:, which are (where
4423:is any of the four
4092:Lagrange resolvents
2202:George Paxton Young
575:Given the equation
9518:Cubic function (3)
9511:Quadratic equation
9241::1:1150–1152 1858.
9095:Harvard University
8922:
8773:
8771:
8401:
8299:
8115:
7925:Francesco Brioschi
7921:elliptic functions
7881:
7700:Francesco Brioschi
7600:
7521:
7483:
7390:
7317:
7276:
7170:
7135:
6957:
6875:
6784:
6704:
6679:
6589:
6514:
6424:
6349:
6262:
6206:
6082:
6019:
5931:
5852:
5773:
5694:
5606:
5545:
5484:
5423:
5346:
5285:
5190:
5165:
5086:
4979:
4925:
4841:
4771:
4701:
4631:
4561:
4481:
4399:
4290:
4237:quadratic equation
4222:
4008:
3871:
3753:
3412:
3248:
3198:
3115:
2980:
2690:
2592:
2416:
2233:Bring–Jerrard form
2183:
2181:
1625:
1623:
1239:Cayley's resolvent
1211:
1209:
790:
670:
528:
360:
158:
39:
9672:
9671:
9613:Quasi-homogeneous
8935:That also yields
8920:
8914:
8678:
8666:
8654:
8636:
8297:
8246:
8082:
8047:
7948:Lagrangian points
7929:Leopold Kronecker
7692:Leopold Kronecker
7664:for real numbers
7520:
7316:
6971:
6970:
6954:
6781:
6703:
6699:
6586:
6421:
6256:
6097:
6096:
5604:
5588:
5543:
5527:
5482:
5466:
5421:
5405:
5277:
5255:
5233:
5217:
5189:
5185:
5084:
5068:
5045:
5022:
4968:
4559:
4394:
4391:
4353:
4123:roots instead of
3996:
3973:
3943:
3865:
3843:
3821:
3799:
3747:
3693:
3642:
3588:
3397:
3367:
3337:
3317:
3193:
3181:
3168:
3166:
3110:
3098:
3088:
3086:
2975:
2906:
2735:The substitution
2683:
2629:
2584:
2520:
2408:
2329:
1205:
1065:
954:
872:
455:Solvable quintics
422:quartic equations
387:quartic equations
195:are members of a
16:(Redirected from
9702:
9535:Quartic equation
9456:
9449:
9442:
9433:
9432:
9390:
9382:
9376:
9368:
9363:. Archived from
9320:
9285:Jörg Bewersdorff
9266::986–992 (1994).
9257:
9230:
9189:
9188:
9176:
9170:
9169:
9157:
9151:
9150:
9138:
9132:
9117:
9111:
9107:David S. Dummit
9105:
9099:
9098:
9090:
9082:
9076:
9070:
9064:
9056::99–130 (1888),
9047:
9041:
9038:
9032:
9024::263-276 (1861)
9015:
9009:
9008:
8998:
8989:
8944:
8931:
8929:
8928:
8923:
8921:
8919:
8913:
8912:
8911:
8898:
8897:
8888:
8887:
8813:
8796:
8782:
8780:
8779:
8774:
8772:
8768:
8767:
8758:
8757:
8738:
8731:
8730:
8718:
8717:
8698:
8688:
8687:
8676:
8664:
8652:
8634:
8633:
8632:
8620:
8619:
8607:
8606:
8584:
8577:
8576:
8561:
8560:
8548:
8547:
8525:
8509:
8508:
8496:
8495:
8473:
8463:
8462:
8450:
8449:
8427:
8410:
8408:
8407:
8402:
8379:
8378:
8363:
8362:
8347:
8346:
8331:
8330:
8308:
8306:
8305:
8300:
8298:
8296:
8295:
8286:
8282:
8281:
8269:
8268:
8252:
8247:
8245:
8244:
8235:
8234:
8233:
8220:
8215:
8214:
8156:angular velocity
8142:, respectively;
8124:
8122:
8121:
8116:
8099:
8098:
8083:
8081:
8080:
8071:
8070:
8069:
8053:
8048:
8046:
8045:
8044:
8022:
8021:
8020:
8004:
7911:
7890:
7888:
7887:
7882:
7873:
7872:
7857:
7856:
7844:
7843:
7834:
7833:
7821:
7820:
7811:
7810:
7798:
7797:
7788:
7787:
7775:
7774:
7754:quartic equation
7729:
7724:septic equations
7721:
7669:
7663:
7609:
7607:
7606:
7601:
7595:
7591:
7586:
7585:
7522:
7513:
7492:
7490:
7489:
7484:
7478:
7477:
7450:
7449:
7437:
7436:
7399:
7397:
7396:
7391:
7385:
7381:
7379:
7378:
7363:
7362:
7350:
7349:
7334:
7333:
7318:
7309:
7285:
7283:
7282:
7277:
7259:
7255:
7244:
7243:
7227:
7226:
7198:
7197:
7179:
7177:
7176:
7171:
7144:
7142:
7141:
7136:
7112:
7111:
7077:
7076:
7061:
7060:
7023:
7022:
6994:
6993:
6967:
6966:
6964:
6963:
6958:
6956:
6955:
6950:
6949:
6948:
6923:
6916:
6911:
6884:
6882:
6881:
6876:
6859:
6858:
6843:
6842:
6827:
6826:
6814:
6813:
6794:
6793:
6791:
6790:
6785:
6783:
6782:
6777:
6776:
6775:
6750:
6743:
6738:
6713:
6711:
6710:
6705:
6701:
6697:
6688:
6686:
6685:
6680:
6663:
6662:
6647:
6646:
6631:
6630:
6618:
6617:
6598:
6596:
6595:
6590:
6588:
6587:
6582:
6581:
6580:
6561:
6554:
6549:
6523:
6521:
6520:
6515:
6498:
6497:
6482:
6481:
6466:
6465:
6453:
6452:
6433:
6431:
6430:
6425:
6423:
6422:
6417:
6416:
6415:
6396:
6389:
6384:
6358:
6356:
6355:
6350:
6336:
6335:
6320:
6319:
6304:
6303:
6291:
6290:
6271:
6269:
6268:
6263:
6261:
6257:
6252:
6241:
6215:
6213:
6212:
6207:
6190:
6189:
6174:
6173:
6158:
6157:
6145:
6144:
6127:
6126:
6121:
6120:
6104:
6091:
6089:
6088:
6083:
6066:
6065:
6050:
6049:
6028:
6026:
6025:
6020:
6000:
5999:
5984:
5983:
5962:
5961:
5940:
5938:
5937:
5932:
5915:
5914:
5899:
5898:
5883:
5882:
5861:
5859:
5858:
5853:
5836:
5835:
5820:
5819:
5804:
5803:
5782:
5780:
5779:
5774:
5757:
5756:
5741:
5740:
5725:
5724:
5703:
5701:
5700:
5695:
5675:
5674:
5659:
5658:
5637:
5636:
5615:
5613:
5612:
5607:
5605:
5597:
5589:
5581:
5576:
5575:
5554:
5552:
5551:
5546:
5544:
5536:
5528:
5520:
5515:
5514:
5493:
5491:
5490:
5485:
5483:
5475:
5467:
5459:
5454:
5453:
5432:
5430:
5429:
5424:
5422:
5414:
5406:
5398:
5393:
5392:
5377:
5376:
5355:
5353:
5352:
5347:
5330:
5329:
5314:
5313:
5294:
5292:
5291:
5286:
5284:
5283:
5278:
5276:
5268:
5262:
5261:
5256:
5254:
5246:
5240:
5239:
5234:
5232:
5224:
5218:
5216:
5208:
5199:
5197:
5196:
5191:
5187:
5183:
5174:
5172:
5171:
5166:
5149:
5148:
5133:
5132:
5117:
5116:
5095:
5093:
5092:
5087:
5085:
5077:
5069:
5061:
5056:
5055:
5046:
5038:
5033:
5032:
5023:
5015:
5010:
5009:
4988:
4986:
4985:
4980:
4969:
4961:
4956:
4955:
4934:
4932:
4931:
4926:
4909:
4908:
4893:
4892:
4877:
4876:
4859:
4858:
4850:
4848:
4847:
4842:
4840:
4839:
4824:
4823:
4814:
4813:
4798:
4797:
4780:
4778:
4777:
4772:
4770:
4769:
4754:
4753:
4744:
4743:
4728:
4727:
4710:
4708:
4707:
4702:
4700:
4699:
4684:
4683:
4674:
4673:
4658:
4657:
4640:
4638:
4637:
4632:
4630:
4629:
4614:
4613:
4604:
4603:
4588:
4587:
4570:
4568:
4567:
4562:
4560:
4555:
4554:
4545:
4540:
4539:
4530:
4529:
4514:
4513:
4490:
4488:
4487:
4482:
4474:
4473:
4458:
4457:
4408:
4406:
4405:
4400:
4395:
4393:
4392:
4390:
4385:
4384:
4375:
4373:
4372:
4359:
4354:
4352:
4347:
4346:
4337:
4335:
4334:
4322:
4321:
4299:
4297:
4296:
4291:
4279:
4278:
4257:
4256:
4231:
4229:
4228:
4223:
4202:
4201:
4186:
4185:
4167:
4166:
4143:
4134:
4128:
4122:
4116:
4111:. However these
4110:
4099:
4089:
4079:
4073:
4067:
4061:
4055:
4048:
4037:
4032:of prime degree
4031:
4017:
4015:
4014:
4009:
3997:
3995:
3994:
3982:
3974:
3972:
3971:
3959:
3954:
3953:
3944:
3936:
3931:
3930:
3915:
3914:
3895:quartic equation
3892:
3880:
3878:
3877:
3872:
3866:
3864:
3859:
3858:
3849:
3844:
3842:
3837:
3836:
3827:
3822:
3820:
3815:
3814:
3805:
3800:
3798:
3793:
3792:
3783:
3762:
3760:
3759:
3754:
3748:
3746:
3741:
3725:
3724:
3699:
3694:
3692:
3687:
3686:
3685:
3648:
3643:
3641:
3636:
3635:
3634:
3594:
3589:
3587:
3582:
3566:
3565:
3543:
3519:
3508:
3507:
3505:
3504:
3501:
3498:
3497:
3496:
3480:
3479:
3478:
3467:
3466:
3465:
3451:
3450:
3449:
3435:
3421:
3419:
3418:
3413:
3408:
3407:
3402:
3398:
3396:
3388:
3378:
3377:
3372:
3368:
3366:
3358:
3348:
3347:
3342:
3338:
3336:
3328:
3318:
3316:
3308:
3284:
3257:
3255:
3254:
3249:
3207:
3205:
3204:
3199:
3194:
3189:
3182:
3177:
3169:
3167:
3162:
3145:
3139:
3124:
3122:
3121:
3116:
3111:
3106:
3099:
3094:
3089:
3087:
3082:
3068:
3065:
3038:
3034:
3030:
3026:
3017:
2989:
2987:
2986:
2981:
2976:
2974:
2973:
2972:
2960:
2959:
2949:
2936:
2935:
2916:
2907:
2905:
2904:
2903:
2891:
2890:
2880:
2867:
2866:
2844:
2826:
2822:
2818:
2803:
2799:
2792:
2785:
2784:
2782:
2781:
2776:
2773:
2761:
2760:
2758:
2757:
2752:
2749:
2731:
2730:
2728:
2727:
2721:
2718:
2699:
2697:
2696:
2691:
2689:
2685:
2684:
2652:
2630:
2622:
2601:
2599:
2598:
2593:
2585:
2583:
2576:
2575:
2565:
2546:
2545:
2529:
2521:
2519:
2512:
2511:
2501:
2482:
2481:
2468:
2463:
2462:
2440:
2434:
2425:
2423:
2422:
2417:
2409:
2407:
2400:
2399:
2389:
2352:
2351:
2338:
2330:
2328:
2321:
2320:
2310:
2291:
2290:
2277:
2272:
2271:
2252:
2230:
2192:
2190:
2189:
2184:
2182:
2175:
2174:
2165:
2164:
2155:
2154:
2136:
2135:
2126:
2125:
2104:
2103:
2085:
2084:
2075:
2074:
2056:
2055:
2040:
2039:
2027:
2026:
2008:
2004:
2003:
1994:
1993:
1984:
1983:
1968:
1967:
1946:
1945:
1936:
1935:
1917:
1916:
1904:
1903:
1888:
1887:
1878:
1877:
1856:
1855:
1834:
1830:
1829:
1820:
1819:
1804:
1803:
1788:
1787:
1778:
1777:
1759:
1758:
1746:
1745:
1721:
1720:
1705:
1704:
1689:
1688:
1679:
1678:
1659:
1634:
1632:
1631:
1626:
1624:
1605:
1604:
1589:
1588:
1567:
1563:
1562:
1547:
1546:
1519:
1518:
1503:
1502:
1493:
1492:
1477:
1476:
1467:
1466:
1448:
1447:
1432:
1431:
1416:
1409:
1408:
1381:
1380:
1365:
1364:
1343:
1342:
1318:
1317:
1293:
1292:
1280:
1279:
1268:
1246:
1241:
1240:
1234:
1220:
1218:
1217:
1212:
1210:
1206:
1204:
1203:
1202:
1189:
1188:
1187:
1169:
1168:
1147:
1146:
1137:
1136:
1115:
1114:
1096:
1095:
1082:
1066:
1064:
1063:
1062:
1049:
1048:
1047:
1029:
1028:
1004:
1003:
985:
984:
971:
955:
953:
952:
951:
938:
937:
936:
903:
902:
889:
873:
871:
870:
869:
856:
855:
854:
829:
799:
797:
796:
791:
768:
767:
752:
751:
736:
735:
716:
715:
713:
712:
706:
703:
679:
677:
676:
671:
645:
644:
629:
628:
613:
612:
597:
596:
551:solvable quintic
537:
535:
534:
529:
509:
508:
489:
443:
369:
367:
366:
361:
337:
336:
321:
320:
305:
304:
289:
288:
265:quintic equation
262:
255:
240:quartic function
214:
201:rational numbers
199:, typically the
194:
190:
186:
182:
178:
174:
167:
165:
164:
159:
138:
137:
122:
121:
106:
105:
90:
89:
47:quintic function
21:
9710:
9709:
9705:
9704:
9703:
9701:
9700:
9699:
9675:
9674:
9673:
9668:
9617:
9556:
9499:Linear equation
9469:
9460:
9419:Bruce Bartlett:
9397:
9370:
9369:
9361:
9308:
9305:Wayback Machine
9245:
9227:
9198:
9193:
9192:
9187:(I): 1150–1152.
9177:
9173:
9158:
9154:
9139:
9135:
9118:
9114:
9106:
9102:
9088:
9083:
9079:
9071:
9067:
9048:
9044:
9039:
9035:
9016:
9012:
8996:
8990:
8986:
8981:
8968:Septic function
8963:Sextic equation
8959:
8952:
8948:
8936:
8915:
8907:
8903:
8899:
8893:
8889:
8886:
8875:
8872:
8871:
8862:
8858:
8854:
8847:
8837:
8830:
8820:
8805:
8803:
8788:
8770:
8769:
8763:
8759:
8753:
8749:
8736:
8735:
8726:
8722:
8713:
8709:
8696:
8695:
8683:
8679:
8628:
8624:
8615:
8611:
8602:
8598:
8582:
8581:
8572:
8568:
8556:
8552:
8543:
8539:
8523:
8522:
8504:
8500:
8491:
8487:
8471:
8470:
8458:
8454:
8445:
8441:
8424:
8422:
8419:
8418:
8374:
8370:
8358:
8354:
8342:
8338:
8326:
8322:
8317:
8314:
8313:
8291:
8287:
8277:
8273:
8264:
8260:
8253:
8251:
8240:
8236:
8229:
8225:
8221:
8219:
8210:
8206:
8204:
8201:
8200:
8186:
8179:
8168:semi-major axis
8141:
8134:
8094:
8090:
8076:
8072:
8065:
8061:
8054:
8052:
8040:
8036:
8023:
8016:
8012:
8005:
8003:
8001:
7998:
7997:
7992:
7981:
7966:
7959:
7944:
7917:Charles Hermite
7898:
7868:
7864:
7852:
7848:
7839:
7835:
7829:
7825:
7816:
7812:
7806:
7802:
7793:
7789:
7783:
7779:
7770:
7766:
7764:
7761:
7760:
7746:
7740:
7727:
7719:
7684:cubic equations
7676:theta functions
7672:Charles Hermite
7665:
7650:
7639:
7637:Beyond radicals
7624:cubic equations
7622:Analogously to
7620:
7581:
7577:
7527:
7523:
7511:
7503:
7500:
7499:
7473:
7469:
7445:
7441:
7432:
7428:
7410:
7407:
7406:
7374:
7370:
7358:
7354:
7345:
7341:
7329:
7325:
7323:
7319:
7307:
7299:
7296:
7295:
7239:
7235:
7222:
7218:
7213:
7209:
7193:
7189:
7187:
7184:
7183:
7153:
7150:
7149:
7107:
7103:
7072:
7068:
7056:
7052:
7018:
7014:
6989:
6985:
6983:
6980:
6979:
6944:
6940:
6924:
6922:
6918:
6912:
6901:
6895:
6892:
6891:
6889:
6854:
6850:
6838:
6834:
6822:
6818:
6809:
6805:
6803:
6800:
6799:
6771:
6767:
6751:
6749:
6745:
6739:
6728:
6722:
6719:
6718:
6716:
6695:
6692:
6691:
6658:
6654:
6642:
6638:
6626:
6622:
6613:
6609:
6607:
6604:
6603:
6576:
6572:
6562:
6560:
6556:
6550:
6539:
6533:
6530:
6529:
6493:
6489:
6477:
6473:
6461:
6457:
6448:
6444:
6442:
6439:
6438:
6411:
6407:
6397:
6395:
6391:
6385:
6374:
6368:
6365:
6364:
6331:
6327:
6315:
6311:
6299:
6295:
6286:
6282:
6280:
6277:
6276:
6242:
6240:
6236:
6225:
6222:
6221:
6185:
6181:
6169:
6165:
6153:
6149:
6140:
6136:
6134:
6131:
6130:
6111:
6110:
6102:
6061:
6057:
6045:
6041:
6039:
6036:
6035:
5995:
5991:
5979:
5975:
5957:
5953:
5951:
5948:
5947:
5910:
5906:
5894:
5890:
5878:
5874:
5872:
5869:
5868:
5831:
5827:
5815:
5811:
5799:
5795:
5793:
5790:
5789:
5752:
5748:
5736:
5732:
5720:
5716:
5714:
5711:
5710:
5670:
5666:
5654:
5650:
5632:
5628:
5626:
5623:
5622:
5596:
5580:
5571:
5567:
5565:
5562:
5561:
5535:
5519:
5510:
5506:
5504:
5501:
5500:
5474:
5458:
5449:
5445:
5443:
5440:
5439:
5413:
5397:
5388:
5384:
5372:
5368:
5366:
5363:
5362:
5325:
5321:
5309:
5305:
5303:
5300:
5299:
5279:
5272:
5267:
5266:
5257:
5250:
5245:
5244:
5235:
5228:
5223:
5222:
5212:
5207:
5205:
5202:
5201:
5181:
5178:
5177:
5144:
5140:
5128:
5124:
5112:
5108:
5106:
5103:
5102:
5076:
5060:
5051:
5047:
5037:
5028:
5024:
5014:
5005:
5001:
4999:
4996:
4995:
4960:
4951:
4947:
4945:
4942:
4941:
4904:
4900:
4888:
4884:
4872:
4868:
4866:
4863:
4862:
4835:
4831:
4819:
4815:
4809:
4805:
4793:
4789:
4787:
4784:
4783:
4765:
4761:
4749:
4745:
4739:
4735:
4723:
4719:
4717:
4714:
4713:
4695:
4691:
4679:
4675:
4669:
4665:
4653:
4649:
4647:
4644:
4643:
4625:
4621:
4609:
4605:
4599:
4595:
4583:
4579:
4577:
4574:
4573:
4550:
4546:
4544:
4535:
4531:
4525:
4521:
4509:
4505:
4503:
4500:
4499:
4469:
4465:
4453:
4449:
4447:
4444:
4443:
4437:
4417:
4386:
4380:
4376:
4374:
4368:
4364:
4363:
4358:
4348:
4342:
4338:
4336:
4330:
4326:
4317:
4313:
4311:
4308:
4307:
4274:
4270:
4252:
4248:
4246:
4243:
4242:
4197:
4193:
4181:
4177:
4162:
4158:
4156:
4153:
4152:
4139:
4130:
4124:
4118:
4112:
4101:
4095:
4085:
4075:
4069:
4063:
4057:
4050:
4039:
4033:
4022:
3990:
3986:
3981:
3967:
3963:
3958:
3949:
3945:
3935:
3926:
3922:
3910:
3906:
3904:
3901:
3900:
3890:
3885:
3860:
3854:
3850:
3848:
3838:
3832:
3828:
3826:
3816:
3810:
3806:
3804:
3794:
3788:
3784:
3782:
3774:
3771:
3770:
3742:
3720:
3716:
3700:
3698:
3688:
3681:
3677:
3649:
3647:
3637:
3630:
3626:
3595:
3593:
3583:
3561:
3557:
3544:
3542:
3528:
3525:
3524:
3514:
3502:
3499:
3494:
3492:
3490:
3489:
3487:
3482:
3476:
3474:
3469:
3460:
3458:
3453:
3444:
3442:
3437:
3426:
3403:
3392:
3387:
3383:
3382:
3373:
3362:
3357:
3353:
3352:
3343:
3332:
3327:
3323:
3322:
3312:
3307:
3293:
3290:
3289:
3263:
3216:
3213:
3212:
3176:
3161:
3144:
3140:
3138:
3136:
3133:
3132:
3093:
3081:
3067:
3066:
3064:
3062:
3059:
3058:
3036:
3032:
3028:
3025:
3019:
3016:
3010:
3008:symmetric group
2996:
2968:
2964:
2955:
2951:
2950:
2931:
2927:
2917:
2915:
2899:
2895:
2886:
2882:
2881:
2862:
2858:
2845:
2843:
2835:
2832:
2831:
2824:
2820:
2805:
2801:
2797:
2787:
2777:
2774:
2771:
2770:
2768:
2763:
2753:
2750:
2744:
2743:
2741:
2736:
2722:
2719:
2712:
2711:
2709:
2704:
2651:
2635:
2631:
2621:
2613:
2610:
2609:
2571:
2567:
2566:
2541:
2537:
2530:
2528:
2507:
2503:
2502:
2477:
2473:
2469:
2467:
2458:
2454:
2452:
2449:
2448:
2436:
2430:
2395:
2391:
2390:
2347:
2343:
2339:
2337:
2316:
2312:
2311:
2286:
2282:
2278:
2276:
2267:
2263:
2261:
2258:
2257:
2247:
2217:
2214:
2180:
2179:
2170:
2166:
2160:
2156:
2150:
2146:
2131:
2127:
2121:
2117:
2099:
2095:
2080:
2076:
2070:
2066:
2051:
2047:
2035:
2031:
2022:
2018:
2006:
2005:
1999:
1995:
1989:
1985:
1979:
1975:
1963:
1959:
1941:
1937:
1931:
1927:
1912:
1908:
1899:
1895:
1883:
1879:
1873:
1869:
1851:
1847:
1832:
1831:
1825:
1821:
1815:
1811:
1799:
1795:
1783:
1779:
1773:
1769:
1754:
1750:
1741:
1737:
1716:
1712:
1700:
1696:
1684:
1680:
1674:
1670:
1660:
1658:
1648:
1646:
1643:
1642:
1622:
1621:
1600:
1596:
1584:
1580:
1565:
1564:
1558:
1554:
1542:
1538:
1514:
1510:
1498:
1494:
1488:
1484:
1472:
1468:
1462:
1458:
1443:
1439:
1427:
1423:
1414:
1413:
1404:
1400:
1376:
1372:
1360:
1356:
1338:
1334:
1313:
1309:
1288:
1284:
1275:
1271:
1269:
1267:
1257:
1255:
1252:
1251:
1244:
1238:
1237:
1225:
1208:
1207:
1198:
1194:
1190:
1183:
1179:
1164:
1160:
1142:
1138:
1132:
1128:
1110:
1106:
1091:
1087:
1083:
1081:
1074:
1068:
1067:
1058:
1054:
1050:
1043:
1039:
1024:
1020:
999:
995:
980:
976:
972:
970:
963:
957:
956:
947:
943:
939:
932:
928:
898:
894:
890:
888:
881:
875:
874:
865:
861:
857:
850:
846:
830:
828:
821:
814:
812:
809:
808:
763:
759:
747:
743:
731:
727:
725:
722:
721:
707:
704:
699:
698:
696:
687:
640:
636:
624:
620:
608:
604:
592:
588:
583:
580:
579:
558:Évariste Galois
504:
500:
498:
495:
494:
464:
457:
434:
399:
332:
328:
316:
312:
300:
296:
284:
280:
275:
272:
271:
257:
246:
228:cubic functions
212:
209:complex numbers
192:
188:
184:
180:
176:
172:
133:
129:
117:
113:
101:
97:
85:
81:
61:
58:
57:
36:critical points
28:
23:
22:
15:
12:
11:
5:
9708:
9698:
9697:
9692:
9687:
9670:
9669:
9667:
9666:
9661:
9656:
9651:
9646:
9641:
9636:
9631:
9625:
9623:
9619:
9618:
9616:
9615:
9610:
9605:
9600:
9595:
9590:
9585:
9580:
9575:
9570:
9564:
9562:
9558:
9557:
9555:
9554:
9549:
9544:
9539:
9538:
9537:
9527:
9526:
9525:
9523:Cubic equation
9515:
9514:
9513:
9503:
9502:
9501:
9491:
9486:
9480:
9478:
9471:
9470:
9459:
9458:
9451:
9444:
9436:
9430:
9429:
9416:
9410:
9404:
9396:
9395:External links
9393:
9392:
9391:
9383:
9359:
9336:
9329:
9322:
9282:
9267:
9242:
9231:
9225:
9210:
9197:
9194:
9191:
9190:
9171:
9152:
9133:
9112:
9100:
9085:Elkies, Noam.
9077:
9075:, p. 207)
9065:
9042:
9033:
9010:
8983:
8982:
8980:
8977:
8976:
8975:
8970:
8965:
8958:
8955:
8950:
8946:
8933:
8932:
8918:
8910:
8906:
8902:
8896:
8892:
8885:
8882:
8879:
8860:
8856:
8852:
8845:
8835:
8828:
8818:
8801:
8785:
8784:
8766:
8762:
8756:
8752:
8748:
8745:
8742:
8739:
8737:
8734:
8729:
8725:
8721:
8716:
8712:
8708:
8705:
8702:
8699:
8697:
8694:
8691:
8686:
8682:
8675:
8672:
8669:
8663:
8660:
8657:
8651:
8648:
8645:
8642:
8639:
8631:
8627:
8623:
8618:
8614:
8610:
8605:
8601:
8597:
8594:
8591:
8588:
8585:
8583:
8580:
8575:
8571:
8567:
8564:
8559:
8555:
8551:
8546:
8542:
8538:
8535:
8532:
8529:
8526:
8524:
8521:
8518:
8515:
8512:
8507:
8503:
8499:
8494:
8490:
8486:
8483:
8480:
8477:
8474:
8472:
8469:
8466:
8461:
8457:
8453:
8448:
8444:
8440:
8437:
8434:
8431:
8428:
8426:
8412:
8411:
8400:
8397:
8394:
8391:
8388:
8385:
8382:
8377:
8373:
8369:
8366:
8361:
8357:
8353:
8350:
8345:
8341:
8337:
8334:
8329:
8325:
8321:
8294:
8290:
8285:
8280:
8276:
8272:
8267:
8263:
8259:
8256:
8250:
8243:
8239:
8232:
8228:
8224:
8218:
8213:
8209:
8184:
8177:
8139:
8132:
8126:
8125:
8114:
8111:
8108:
8105:
8102:
8097:
8093:
8089:
8086:
8079:
8075:
8068:
8064:
8060:
8057:
8051:
8043:
8039:
8035:
8032:
8029:
8026:
8019:
8015:
8011:
8008:
7990:
7979:
7964:
7957:
7943:
7940:
7892:
7891:
7879:
7876:
7871:
7867:
7863:
7860:
7855:
7851:
7847:
7842:
7838:
7832:
7828:
7824:
7819:
7815:
7809:
7805:
7801:
7796:
7792:
7786:
7782:
7778:
7773:
7769:
7742:Main article:
7739:
7736:
7638:
7635:
7619:
7614:
7613:
7612:
7611:
7610:
7599:
7594:
7589:
7584:
7580:
7576:
7573:
7570:
7567:
7564:
7561:
7558:
7555:
7552:
7549:
7546:
7543:
7540:
7537:
7534:
7531:
7526:
7519:
7516:
7510:
7507:
7495:
7494:
7493:
7482:
7476:
7472:
7468:
7465:
7462:
7459:
7456:
7453:
7448:
7444:
7440:
7435:
7431:
7427:
7424:
7420:
7417:
7414:
7402:
7401:
7400:
7389:
7384:
7377:
7373:
7369:
7366:
7361:
7357:
7353:
7348:
7344:
7340:
7337:
7332:
7328:
7322:
7315:
7312:
7306:
7303:
7287:
7286:
7275:
7272:
7269:
7265:
7262:
7258:
7254:
7250:
7247:
7242:
7238:
7233:
7230:
7225:
7221:
7216:
7212:
7208:
7204:
7201:
7196:
7192:
7169:
7166:
7163:
7160:
7157:
7146:
7145:
7134:
7131:
7128:
7125:
7122:
7118:
7115:
7110:
7106:
7101:
7098:
7095:
7092:
7089:
7086:
7083:
7080:
7075:
7071:
7067:
7064:
7059:
7055:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7021:
7017:
7012:
7009:
7006:
7003:
7000:
6997:
6992:
6988:
6973:
6972:
6969:
6968:
6953:
6947:
6943:
6939:
6936:
6933:
6930:
6927:
6921:
6915:
6910:
6907:
6904:
6900:
6887:
6885:
6874:
6871:
6868:
6865:
6862:
6857:
6853:
6849:
6846:
6841:
6837:
6833:
6830:
6825:
6821:
6817:
6812:
6808:
6796:
6795:
6780:
6774:
6770:
6766:
6763:
6760:
6757:
6754:
6748:
6742:
6737:
6734:
6731:
6727:
6714:
6689:
6678:
6675:
6672:
6669:
6666:
6661:
6657:
6653:
6650:
6645:
6641:
6637:
6634:
6629:
6625:
6621:
6616:
6612:
6600:
6599:
6585:
6579:
6575:
6571:
6568:
6565:
6559:
6553:
6548:
6545:
6542:
6538:
6526:
6524:
6513:
6510:
6507:
6504:
6501:
6496:
6492:
6488:
6485:
6480:
6476:
6472:
6469:
6464:
6460:
6456:
6451:
6447:
6435:
6434:
6420:
6414:
6410:
6406:
6403:
6400:
6394:
6388:
6383:
6380:
6377:
6373:
6361:
6359:
6348:
6345:
6342:
6339:
6334:
6330:
6326:
6323:
6318:
6314:
6310:
6307:
6302:
6298:
6294:
6289:
6285:
6273:
6272:
6260:
6255:
6251:
6248:
6245:
6239:
6235:
6232:
6229:
6218:
6216:
6205:
6202:
6199:
6196:
6193:
6188:
6184:
6180:
6177:
6172:
6168:
6164:
6161:
6156:
6152:
6148:
6143:
6139:
6107:roots of unity
6099:
6098:
6095:
6094:
6092:
6081:
6078:
6075:
6072:
6069:
6064:
6060:
6056:
6053:
6048:
6044:
6032:
6031:
6029:
6018:
6015:
6012:
6009:
6006:
6003:
5998:
5994:
5990:
5987:
5982:
5978:
5974:
5971:
5968:
5965:
5960:
5956:
5944:
5943:
5941:
5930:
5927:
5924:
5921:
5918:
5913:
5909:
5905:
5902:
5897:
5893:
5889:
5886:
5881:
5877:
5865:
5864:
5862:
5851:
5848:
5845:
5842:
5839:
5834:
5830:
5826:
5823:
5818:
5814:
5810:
5807:
5802:
5798:
5786:
5785:
5783:
5772:
5769:
5766:
5763:
5760:
5755:
5751:
5747:
5744:
5739:
5735:
5731:
5728:
5723:
5719:
5707:
5706:
5704:
5693:
5690:
5687:
5684:
5681:
5678:
5673:
5669:
5665:
5662:
5657:
5653:
5649:
5646:
5643:
5640:
5635:
5631:
5619:
5618:
5616:
5603:
5600:
5595:
5592:
5587:
5584:
5579:
5574:
5570:
5558:
5557:
5555:
5542:
5539:
5534:
5531:
5526:
5523:
5518:
5513:
5509:
5497:
5496:
5494:
5481:
5478:
5473:
5470:
5465:
5462:
5457:
5452:
5448:
5436:
5435:
5433:
5420:
5417:
5412:
5409:
5404:
5401:
5396:
5391:
5387:
5383:
5380:
5375:
5371:
5359:
5358:
5356:
5345:
5342:
5339:
5336:
5333:
5328:
5324:
5320:
5317:
5312:
5308:
5296:
5295:
5282:
5275:
5271:
5265:
5260:
5253:
5249:
5243:
5238:
5231:
5227:
5221:
5215:
5211:
5175:
5164:
5161:
5158:
5155:
5152:
5147:
5143:
5139:
5136:
5131:
5127:
5123:
5120:
5115:
5111:
5099:
5098:
5096:
5083:
5080:
5075:
5072:
5067:
5064:
5059:
5054:
5050:
5044:
5041:
5036:
5031:
5027:
5021:
5018:
5013:
5008:
5004:
4992:
4991:
4989:
4978:
4975:
4972:
4967:
4964:
4959:
4954:
4950:
4938:
4937:
4935:
4924:
4921:
4918:
4915:
4912:
4907:
4903:
4899:
4896:
4891:
4887:
4883:
4880:
4875:
4871:
4852:
4851:
4838:
4834:
4830:
4827:
4822:
4818:
4812:
4808:
4804:
4801:
4796:
4792:
4781:
4768:
4764:
4760:
4757:
4752:
4748:
4742:
4738:
4734:
4731:
4726:
4722:
4711:
4698:
4694:
4690:
4687:
4682:
4678:
4672:
4668:
4664:
4661:
4656:
4652:
4641:
4628:
4624:
4620:
4617:
4612:
4608:
4602:
4598:
4594:
4591:
4586:
4582:
4571:
4558:
4553:
4549:
4543:
4538:
4534:
4528:
4524:
4520:
4517:
4512:
4508:
4480:
4477:
4472:
4468:
4464:
4461:
4456:
4452:
4436:
4433:
4415:
4410:
4409:
4398:
4389:
4383:
4379:
4371:
4367:
4362:
4357:
4351:
4345:
4341:
4333:
4329:
4325:
4320:
4316:
4301:
4300:
4289:
4285:
4282:
4277:
4273:
4269:
4266:
4263:
4260:
4255:
4251:
4233:
4232:
4221:
4217:
4214:
4211:
4208:
4205:
4200:
4196:
4192:
4189:
4184:
4180:
4176:
4173:
4170:
4165:
4161:
4062:is the sum of
4019:
4018:
4007:
4003:
4000:
3993:
3989:
3985:
3980:
3977:
3970:
3966:
3962:
3957:
3952:
3948:
3942:
3939:
3934:
3929:
3925:
3921:
3918:
3913:
3909:
3888:
3882:
3881:
3870:
3863:
3857:
3853:
3847:
3841:
3835:
3831:
3825:
3819:
3813:
3809:
3803:
3797:
3791:
3787:
3781:
3778:
3764:
3763:
3752:
3745:
3740:
3737:
3734:
3731:
3728:
3723:
3719:
3715:
3712:
3709:
3706:
3703:
3697:
3691:
3684:
3680:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3646:
3640:
3633:
3629:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3592:
3586:
3581:
3578:
3575:
3572:
3569:
3564:
3560:
3556:
3553:
3550:
3547:
3541:
3538:
3535:
3532:
3423:
3422:
3411:
3406:
3401:
3395:
3391:
3386:
3381:
3376:
3371:
3365:
3361:
3356:
3351:
3346:
3341:
3335:
3331:
3326:
3321:
3315:
3311:
3306:
3303:
3300:
3297:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3209:
3208:
3197:
3192:
3188:
3185:
3180:
3175:
3172:
3165:
3160:
3157:
3154:
3151:
3148:
3143:
3126:
3125:
3114:
3109:
3105:
3102:
3097:
3092:
3085:
3080:
3077:
3074:
3071:
3023:
3014:
3004:solvable group
2995:
2992:
2991:
2990:
2979:
2971:
2967:
2963:
2958:
2954:
2948:
2945:
2942:
2939:
2934:
2930:
2926:
2923:
2920:
2914:
2911:
2902:
2898:
2894:
2889:
2885:
2879:
2876:
2873:
2870:
2865:
2861:
2857:
2854:
2851:
2848:
2842:
2839:
2701:
2700:
2688:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2650:
2647:
2644:
2641:
2638:
2634:
2628:
2625:
2620:
2617:
2603:
2602:
2591:
2588:
2582:
2579:
2574:
2570:
2564:
2561:
2558:
2555:
2552:
2549:
2544:
2540:
2536:
2533:
2527:
2524:
2518:
2515:
2510:
2506:
2500:
2497:
2494:
2491:
2488:
2485:
2480:
2476:
2472:
2466:
2461:
2457:
2441:are rational.
2427:
2426:
2415:
2412:
2406:
2403:
2398:
2394:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2350:
2346:
2342:
2336:
2333:
2327:
2324:
2319:
2315:
2309:
2306:
2303:
2300:
2297:
2294:
2289:
2285:
2281:
2275:
2270:
2266:
2213:
2210:
2194:
2193:
2178:
2173:
2169:
2163:
2159:
2153:
2149:
2145:
2142:
2139:
2134:
2130:
2124:
2120:
2116:
2113:
2110:
2107:
2102:
2098:
2094:
2091:
2088:
2083:
2079:
2073:
2069:
2065:
2062:
2059:
2054:
2050:
2046:
2043:
2038:
2034:
2030:
2025:
2021:
2017:
2014:
2011:
2009:
2007:
2002:
1998:
1992:
1988:
1982:
1978:
1974:
1971:
1966:
1962:
1958:
1955:
1952:
1949:
1944:
1940:
1934:
1930:
1926:
1923:
1920:
1915:
1911:
1907:
1902:
1898:
1894:
1891:
1886:
1882:
1876:
1872:
1868:
1865:
1862:
1859:
1854:
1850:
1846:
1843:
1840:
1837:
1835:
1833:
1828:
1824:
1818:
1814:
1810:
1807:
1802:
1798:
1794:
1791:
1786:
1782:
1776:
1772:
1768:
1765:
1762:
1757:
1753:
1749:
1744:
1740:
1736:
1733:
1730:
1727:
1724:
1719:
1715:
1711:
1708:
1703:
1699:
1695:
1692:
1687:
1683:
1677:
1673:
1669:
1666:
1663:
1661:
1657:
1654:
1651:
1650:
1636:
1635:
1620:
1617:
1614:
1611:
1608:
1603:
1599:
1595:
1592:
1587:
1583:
1579:
1576:
1573:
1570:
1568:
1566:
1561:
1557:
1553:
1550:
1545:
1541:
1537:
1534:
1531:
1528:
1525:
1522:
1517:
1513:
1509:
1506:
1501:
1497:
1491:
1487:
1483:
1480:
1475:
1471:
1465:
1461:
1457:
1454:
1451:
1446:
1442:
1438:
1435:
1430:
1426:
1422:
1419:
1417:
1415:
1412:
1407:
1403:
1399:
1396:
1393:
1390:
1387:
1384:
1379:
1375:
1371:
1368:
1363:
1359:
1355:
1352:
1349:
1346:
1341:
1337:
1333:
1330:
1327:
1324:
1321:
1316:
1312:
1308:
1305:
1302:
1299:
1296:
1291:
1287:
1283:
1278:
1274:
1270:
1266:
1263:
1260:
1259:
1222:
1221:
1201:
1197:
1193:
1186:
1182:
1178:
1175:
1172:
1167:
1163:
1159:
1156:
1153:
1150:
1145:
1141:
1135:
1131:
1127:
1124:
1121:
1118:
1113:
1109:
1105:
1102:
1099:
1094:
1090:
1086:
1080:
1077:
1075:
1073:
1070:
1069:
1061:
1057:
1053:
1046:
1042:
1038:
1035:
1032:
1027:
1023:
1019:
1016:
1013:
1010:
1007:
1002:
998:
994:
991:
988:
983:
979:
975:
969:
966:
964:
962:
959:
958:
950:
946:
942:
935:
931:
927:
924:
921:
918:
915:
912:
909:
906:
901:
897:
893:
887:
884:
882:
880:
877:
876:
868:
864:
860:
853:
849:
845:
842:
839:
836:
833:
827:
824:
822:
820:
817:
816:
802:
801:
789:
786:
783:
780:
777:
774:
771:
766:
762:
758:
755:
750:
746:
742:
739:
734:
730:
681:
680:
669:
666:
663:
660:
657:
654:
651:
648:
643:
639:
635:
632:
627:
623:
619:
616:
611:
607:
603:
600:
595:
591:
587:
539:
538:
527:
524:
521:
518:
515:
512:
507:
503:
456:
453:
398:
395:
371:
370:
358:
355:
352:
349:
346:
343:
340:
335:
331:
327:
324:
319:
315:
311:
308:
303:
299:
295:
292:
287:
283:
279:
169:
168:
156:
153:
150:
147:
144:
141:
136:
132:
128:
125:
120:
116:
112:
109:
104:
100:
96:
93:
88:
84:
80:
77:
74:
71:
68:
65:
26:
9:
6:
4:
3:
2:
9707:
9696:
9693:
9691:
9690:Galois theory
9688:
9686:
9683:
9682:
9680:
9665:
9664:Gröbner basis
9662:
9660:
9657:
9655:
9652:
9650:
9647:
9645:
9642:
9640:
9637:
9635:
9632:
9630:
9629:Factorization
9627:
9626:
9624:
9620:
9614:
9611:
9609:
9606:
9604:
9601:
9599:
9596:
9594:
9591:
9589:
9586:
9584:
9581:
9579:
9576:
9574:
9571:
9569:
9566:
9565:
9563:
9561:By properties
9559:
9553:
9550:
9548:
9545:
9543:
9540:
9536:
9533:
9532:
9531:
9528:
9524:
9521:
9520:
9519:
9516:
9512:
9509:
9508:
9507:
9504:
9500:
9497:
9496:
9495:
9492:
9490:
9487:
9485:
9482:
9481:
9479:
9477:
9472:
9468:
9464:
9457:
9452:
9450:
9445:
9443:
9438:
9437:
9434:
9427:
9423:
9422:
9417:
9414:
9411:
9408:
9405:
9402:
9399:
9398:
9389:
9384:
9380:
9374:
9366:
9362:
9360:3-540-43826-2
9356:
9352:
9351:
9346:
9342:
9337:
9334:
9330:
9327:
9323:
9319:
9315:
9311:
9306:
9302:
9299:. Chapter 8 (
9298:
9297:0-8218-3817-2
9294:
9290:
9286:
9283:
9280:
9279:0-412-34550-1
9276:
9272:
9271:Galois Theory
9269:Ian Stewart,
9268:
9265:
9261:
9256:
9252:
9248:
9243:
9240:
9236:
9232:
9228:
9226:0-486-49528-0
9222:
9218:
9217:
9211:
9208:
9204:
9200:
9199:
9186:
9182:
9175:
9167:
9163:
9156:
9149:(I): 508–515.
9148:
9144:
9137:
9130:
9126:
9122:
9116:
9110:
9104:
9096:
9092:
9089:a x + b x + c
9081:
9074:
9069:
9063:
9059:
9055:
9052:
9046:
9037:
9031:
9027:
9023:
9020:
9014:
9007:(3): 282–286.
9006:
9002:
8995:
8988:
8984:
8974:
8971:
8969:
8966:
8964:
8961:
8960:
8954:
8943:
8939:
8916:
8908:
8904:
8900:
8894:
8890:
8883:
8880:
8877:
8870:
8869:
8868:
8866:
8851:
8844:
8839:
8834:
8827:
8824:
8817:
8812:
8809:= 1.491 x 10
8808:
8800:
8795:
8792:= 1.501 x 10
8791:
8764:
8760:
8754:
8750:
8746:
8743:
8740:
8732:
8727:
8723:
8719:
8714:
8710:
8706:
8703:
8700:
8692:
8684:
8680:
8673:
8670:
8667:
8661:
8658:
8655:
8649:
8646:
8643:
8640:
8629:
8625:
8616:
8612:
8608:
8603:
8599:
8592:
8589:
8586:
8578:
8573:
8569:
8565:
8557:
8553:
8549:
8544:
8540:
8533:
8530:
8527:
8519:
8516:
8513:
8505:
8501:
8497:
8492:
8488:
8481:
8478:
8475:
8467:
8459:
8455:
8451:
8446:
8442:
8435:
8432:
8429:
8417:
8416:
8415:
8398:
8395:
8392:
8389:
8386:
8383:
8380:
8375:
8371:
8367:
8364:
8359:
8355:
8351:
8348:
8343:
8339:
8335:
8332:
8327:
8323:
8319:
8312:
8311:
8310:
8292:
8288:
8278:
8274:
8270:
8265:
8261:
8254:
8248:
8241:
8237:
8230:
8226:
8222:
8216:
8211:
8207:
8197:
8195:
8191:
8187:
8180:
8173:
8169:
8165:
8161:
8157:
8153:
8149:
8145:
8138:
8131:
8109:
8106:
8103:
8095:
8091:
8087:
8084:
8077:
8073:
8066:
8062:
8058:
8055:
8049:
8041:
8033:
8030:
8027:
8017:
8013:
8009:
8006:
7996:
7995:
7994:
7989:
7985:
7978:
7974:
7970:
7963:
7956:
7951:
7949:
7939:
7937:
7936:Bring radical
7932:
7930:
7926:
7922:
7918:
7913:
7909:
7905:
7901:
7897:
7877:
7874:
7869:
7865:
7861:
7858:
7853:
7849:
7845:
7840:
7836:
7830:
7826:
7822:
7817:
7813:
7807:
7803:
7799:
7794:
7790:
7784:
7780:
7776:
7771:
7767:
7759:
7758:
7757:
7755:
7751:
7745:
7744:Bring radical
7735:
7733:
7725:
7717:
7713:
7712:Galois theory
7709:
7705:
7701:
7697:
7693:
7689:
7685:
7681:
7677:
7673:
7668:
7661:
7657:
7653:
7648:
7647:ultraradicals
7644:
7634:
7631:
7630:
7625:
7618:
7597:
7592:
7587:
7582:
7578:
7574:
7568:
7565:
7562:
7559:
7553:
7550:
7544:
7541:
7538:
7535:
7529:
7524:
7517:
7514:
7508:
7505:
7498:
7497:
7496:
7480:
7474:
7470:
7466:
7463:
7460:
7457:
7454:
7446:
7442:
7438:
7433:
7429:
7425:
7418:
7415:
7412:
7405:
7404:
7403:
7387:
7382:
7375:
7371:
7367:
7359:
7355:
7351:
7346:
7342:
7338:
7330:
7326:
7320:
7313:
7310:
7304:
7301:
7294:
7293:
7292:
7291:
7290:
7273:
7270:
7267:
7263:
7260:
7256:
7252:
7248:
7245:
7240:
7236:
7231:
7228:
7223:
7219:
7214:
7210:
7206:
7202:
7199:
7194:
7190:
7182:
7181:
7180:
7167:
7164:
7161:
7158:
7155:
7132:
7129:
7126:
7123:
7120:
7116:
7113:
7108:
7104:
7096:
7093:
7090:
7087:
7084:
7081:
7078:
7073:
7069:
7062:
7057:
7053:
7045:
7042:
7039:
7036:
7033:
7030:
7024:
7019:
7015:
7007:
7004:
7001:
6995:
6990:
6986:
6978:
6977:
6976:
6951:
6945:
6937:
6931:
6928:
6925:
6919:
6913:
6908:
6905:
6902:
6898:
6888:
6886:
6872:
6869:
6866:
6863:
6860:
6855:
6851:
6847:
6844:
6839:
6835:
6831:
6828:
6823:
6819:
6815:
6810:
6806:
6798:
6797:
6778:
6772:
6764:
6758:
6755:
6752:
6746:
6740:
6735:
6732:
6729:
6725:
6715:
6690:
6676:
6673:
6670:
6667:
6664:
6659:
6655:
6651:
6648:
6643:
6639:
6635:
6632:
6627:
6623:
6619:
6614:
6610:
6602:
6601:
6583:
6577:
6573:
6569:
6566:
6563:
6557:
6551:
6546:
6543:
6540:
6536:
6527:
6525:
6511:
6508:
6505:
6502:
6499:
6494:
6490:
6486:
6483:
6478:
6474:
6470:
6467:
6462:
6458:
6454:
6449:
6445:
6437:
6436:
6418:
6412:
6408:
6404:
6401:
6398:
6392:
6386:
6381:
6378:
6375:
6371:
6362:
6360:
6346:
6343:
6340:
6337:
6332:
6328:
6324:
6321:
6316:
6312:
6308:
6305:
6300:
6296:
6292:
6287:
6283:
6275:
6274:
6258:
6253:
6249:
6246:
6243:
6237:
6233:
6230:
6227:
6219:
6217:
6203:
6200:
6197:
6194:
6191:
6186:
6182:
6178:
6175:
6170:
6166:
6162:
6159:
6154:
6150:
6146:
6141:
6137:
6129:
6128:
6125:
6124:
6123:
6118:
6114:
6108:
6093:
6079:
6076:
6073:
6070:
6067:
6062:
6058:
6054:
6051:
6046:
6042:
6034:
6033:
6030:
6013:
6010:
6007:
6004:
6001:
5996:
5992:
5988:
5985:
5980:
5976:
5972:
5966:
5963:
5958:
5954:
5946:
5945:
5942:
5928:
5925:
5922:
5919:
5916:
5911:
5907:
5903:
5900:
5895:
5891:
5887:
5884:
5879:
5875:
5867:
5866:
5863:
5849:
5846:
5843:
5840:
5837:
5832:
5828:
5824:
5821:
5816:
5812:
5808:
5805:
5800:
5796:
5788:
5787:
5784:
5770:
5767:
5764:
5761:
5758:
5753:
5749:
5745:
5742:
5737:
5733:
5729:
5726:
5721:
5717:
5709:
5708:
5705:
5688:
5685:
5682:
5679:
5676:
5671:
5667:
5663:
5660:
5655:
5651:
5647:
5641:
5638:
5633:
5629:
5621:
5620:
5617:
5601:
5598:
5593:
5590:
5585:
5582:
5577:
5572:
5568:
5560:
5559:
5556:
5540:
5537:
5532:
5529:
5524:
5521:
5516:
5511:
5507:
5499:
5498:
5495:
5479:
5476:
5471:
5468:
5463:
5460:
5455:
5450:
5446:
5438:
5437:
5434:
5418:
5415:
5410:
5407:
5402:
5399:
5394:
5389:
5385:
5381:
5378:
5373:
5369:
5361:
5360:
5357:
5343:
5340:
5337:
5334:
5331:
5326:
5322:
5318:
5315:
5310:
5306:
5298:
5297:
5280:
5273:
5269:
5263:
5258:
5251:
5247:
5241:
5236:
5229:
5225:
5219:
5213:
5209:
5176:
5162:
5159:
5156:
5153:
5150:
5145:
5141:
5137:
5134:
5129:
5125:
5121:
5118:
5113:
5109:
5101:
5100:
5097:
5081:
5078:
5073:
5070:
5065:
5062:
5057:
5052:
5048:
5042:
5039:
5034:
5029:
5025:
5019:
5016:
5011:
5006:
5002:
4994:
4993:
4990:
4976:
4973:
4970:
4965:
4962:
4957:
4952:
4948:
4940:
4939:
4936:
4922:
4919:
4916:
4913:
4910:
4905:
4901:
4897:
4894:
4889:
4885:
4881:
4878:
4873:
4869:
4861:
4860:
4857:
4856:
4855:
4836:
4832:
4828:
4825:
4820:
4816:
4810:
4806:
4802:
4799:
4794:
4790:
4782:
4766:
4762:
4758:
4755:
4750:
4746:
4740:
4736:
4732:
4729:
4724:
4720:
4712:
4696:
4692:
4688:
4685:
4680:
4676:
4670:
4666:
4662:
4659:
4654:
4650:
4642:
4626:
4622:
4618:
4615:
4610:
4606:
4600:
4596:
4592:
4589:
4584:
4580:
4572:
4556:
4551:
4547:
4541:
4536:
4532:
4526:
4522:
4518:
4515:
4510:
4506:
4498:
4497:
4496:
4494:
4478:
4475:
4470:
4466:
4462:
4459:
4454:
4450:
4440:
4432:
4430:
4426:
4422:
4418:
4396:
4387:
4381:
4377:
4369:
4365:
4360:
4355:
4349:
4343:
4339:
4331:
4327:
4323:
4318:
4314:
4306:
4305:
4304:
4287:
4283:
4280:
4275:
4271:
4267:
4264:
4261:
4258:
4253:
4249:
4241:
4240:
4239:
4238:
4219:
4215:
4212:
4209:
4206:
4203:
4198:
4194:
4190:
4187:
4182:
4178:
4174:
4171:
4168:
4163:
4159:
4151:
4150:
4149:
4147:
4142:
4136:
4133:
4127:
4121:
4115:
4108:
4104:
4098:
4093:
4088:
4083:
4078:
4072:
4066:
4060:
4053:
4046:
4042:
4036:
4029:
4025:
4005:
4001:
3998:
3991:
3987:
3983:
3978:
3975:
3968:
3964:
3960:
3955:
3950:
3946:
3940:
3937:
3932:
3927:
3923:
3919:
3916:
3911:
3907:
3899:
3898:
3897:
3896:
3891:
3868:
3861:
3855:
3851:
3845:
3839:
3833:
3829:
3823:
3817:
3811:
3807:
3801:
3795:
3789:
3785:
3779:
3776:
3769:
3768:
3767:
3750:
3743:
3735:
3732:
3729:
3721:
3713:
3710:
3707:
3704:
3695:
3689:
3682:
3674:
3671:
3668:
3659:
3656:
3653:
3644:
3638:
3631:
3623:
3620:
3617:
3608:
3605:
3602:
3599:
3590:
3584:
3576:
3573:
3570:
3562:
3554:
3551:
3548:
3539:
3536:
3533:
3530:
3523:
3522:
3521:
3518:= −1.84208...
3517:
3512:
3485:
3472:
3464:
3456:
3448:
3440:
3433:
3429:
3409:
3404:
3399:
3393:
3389:
3384:
3379:
3374:
3369:
3363:
3359:
3354:
3349:
3344:
3339:
3333:
3329:
3324:
3319:
3313:
3309:
3304:
3301:
3298:
3295:
3288:
3287:
3286:
3282:
3278:
3274:
3270:
3266:
3259:
3242:
3239:
3236:
3233:
3227:
3224:
3221:
3218:
3195:
3190:
3186:
3183:
3178:
3173:
3170:
3163:
3158:
3155:
3152:
3149:
3146:
3141:
3131:
3130:
3129:
3112:
3107:
3103:
3100:
3095:
3090:
3083:
3078:
3075:
3072:
3069:
3057:
3056:
3055:
3053:
3049:
3045:
3040:
3022:
3013:
3009:
3005:
3001:
2977:
2969:
2965:
2961:
2956:
2952:
2943:
2940:
2937:
2932:
2928:
2924:
2918:
2912:
2909:
2900:
2896:
2892:
2887:
2883:
2874:
2871:
2868:
2863:
2859:
2855:
2849:
2846:
2840:
2837:
2830:
2829:
2828:
2816:
2812:
2808:
2794:
2790:
2780:
2766:
2756:
2748:
2739:
2733:
2725:
2716:
2707:
2686:
2677:
2674:
2671:
2662:
2659:
2656:
2648:
2645:
2642:
2639:
2636:
2632:
2626:
2623:
2618:
2615:
2608:
2607:
2606:
2589:
2586:
2580:
2577:
2572:
2568:
2559:
2556:
2553:
2550:
2542:
2538:
2534:
2531:
2525:
2522:
2516:
2513:
2508:
2504:
2495:
2492:
2489:
2486:
2478:
2474:
2470:
2464:
2459:
2455:
2447:
2446:
2445:
2442:
2439:
2433:
2413:
2410:
2404:
2401:
2396:
2392:
2383:
2380:
2377:
2374:
2365:
2362:
2359:
2356:
2348:
2344:
2340:
2334:
2331:
2325:
2322:
2317:
2313:
2304:
2301:
2298:
2295:
2287:
2283:
2279:
2273:
2268:
2264:
2256:
2255:
2254:
2250:
2245:
2241:
2236:
2234:
2231:, called the
2228:
2224:
2220:
2209:
2207:
2206:Daniel Lazard
2203:
2198:
2176:
2171:
2167:
2161:
2157:
2151:
2147:
2143:
2140:
2137:
2132:
2128:
2122:
2118:
2114:
2111:
2108:
2105:
2100:
2096:
2092:
2089:
2086:
2081:
2077:
2071:
2067:
2063:
2060:
2057:
2052:
2048:
2044:
2041:
2036:
2032:
2028:
2023:
2019:
2015:
2012:
2010:
2000:
1996:
1990:
1986:
1980:
1976:
1972:
1969:
1964:
1960:
1956:
1953:
1950:
1947:
1942:
1938:
1932:
1928:
1924:
1921:
1918:
1913:
1909:
1905:
1900:
1896:
1892:
1889:
1884:
1880:
1874:
1870:
1866:
1863:
1860:
1857:
1852:
1848:
1844:
1841:
1838:
1836:
1826:
1822:
1816:
1812:
1808:
1805:
1800:
1796:
1792:
1789:
1784:
1780:
1774:
1770:
1766:
1763:
1760:
1755:
1751:
1747:
1742:
1738:
1734:
1731:
1728:
1725:
1722:
1717:
1713:
1709:
1706:
1701:
1697:
1693:
1690:
1685:
1681:
1675:
1671:
1667:
1664:
1662:
1655:
1641:
1640:
1639:
1618:
1615:
1612:
1609:
1606:
1601:
1597:
1593:
1590:
1585:
1581:
1577:
1574:
1571:
1569:
1559:
1555:
1551:
1548:
1543:
1539:
1535:
1532:
1529:
1526:
1523:
1520:
1515:
1511:
1507:
1504:
1499:
1495:
1489:
1485:
1481:
1478:
1473:
1469:
1463:
1459:
1455:
1452:
1449:
1444:
1440:
1436:
1433:
1428:
1424:
1420:
1418:
1405:
1401:
1397:
1394:
1391:
1388:
1385:
1382:
1377:
1373:
1369:
1366:
1361:
1357:
1353:
1350:
1347:
1344:
1339:
1335:
1331:
1325:
1322:
1314:
1310:
1306:
1303:
1300:
1297:
1289:
1285:
1281:
1276:
1272:
1264:
1261:
1250:
1249:
1248:
1242:
1232:
1228:
1199:
1195:
1191:
1184:
1180:
1176:
1173:
1170:
1165:
1161:
1157:
1154:
1151:
1148:
1143:
1139:
1133:
1129:
1125:
1122:
1119:
1116:
1111:
1107:
1103:
1100:
1097:
1092:
1088:
1084:
1078:
1076:
1071:
1059:
1055:
1051:
1044:
1040:
1036:
1033:
1030:
1025:
1021:
1017:
1014:
1011:
1008:
1005:
1000:
996:
992:
989:
986:
981:
977:
973:
967:
965:
960:
948:
944:
940:
933:
929:
925:
922:
919:
916:
913:
910:
907:
904:
899:
895:
891:
885:
883:
878:
866:
862:
858:
851:
847:
843:
840:
837:
834:
831:
825:
823:
818:
807:
806:
805:
787:
784:
781:
778:
775:
772:
769:
764:
760:
756:
753:
748:
744:
740:
737:
732:
728:
720:
719:
718:
711:
702:
694:
690:
686:
667:
664:
661:
658:
655:
652:
649:
646:
641:
637:
633:
630:
625:
621:
617:
614:
609:
605:
601:
598:
593:
589:
585:
578:
577:
576:
573:
571:
570:Arthur Cayley
567:
566:Galois theory
563:
559:
554:
552:
546:
544:
525:
522:
519:
516:
513:
510:
505:
501:
493:
492:
491:
487:
483:
479:
475:
471:
467:
462:
452:
450:
445:
441:
437:
432:
428:
423:
419:
415:
411:
406:
404:
394:
392:
388:
384:
380:
376:
356:
353:
350:
347:
344:
341:
338:
333:
329:
325:
322:
317:
313:
309:
306:
301:
297:
293:
290:
285:
281:
277:
270:
269:
268:
267:of the form:
266:
260:
256:and assuming
253:
249:
243:
241:
237:
233:
232:local maximum
229:
224:
222:
218:
210:
206:
202:
198:
154:
151:
148:
145:
142:
139:
134:
130:
126:
123:
118:
114:
110:
107:
102:
98:
94:
91:
86:
82:
78:
75:
69:
63:
56:
55:
54:
52:
48:
44:
37:
32:
19:
9659:Discriminant
9578:Multivariate
9541:
9428:(April 2024)
9420:
9387:
9365:the original
9349:
9332:
9325:
9317:
9313:
9309:
9288:
9270:
9263:
9259:
9254:
9250:
9246:
9238:
9234:
9215:
9206:
9202:
9184:
9180:
9174:
9165:
9161:
9155:
9146:
9142:
9136:
9115:
9103:
9087:"Trinomials
9080:
9073:Lazard (2004
9068:
9053:
9050:
9045:
9036:
9021:
9018:
9013:
9004:
9000:
8987:
8941:
8937:
8934:
8867:, given by:
8849:
8842:
8840:
8832:
8825:
8815:
8810:
8806:
8798:
8793:
8789:
8786:
8413:
8198:
8182:
8175:
8171:
8163:
8159:
8151:
8143:
8136:
8129:
8127:
7987:
7976:
7961:
7954:
7952:
7945:
7933:
7914:
7907:
7903:
7899:
7893:
7747:
7696:group theory
7686:by means of
7666:
7659:
7655:
7651:
7641:About 1835,
7640:
7627:
7621:
7616:
7288:
7147:
6974:
6116:
6112:
6100:
4853:
4492:
4441:
4438:
4420:
4413:
4411:
4302:
4234:
4140:
4137:
4131:
4125:
4119:
4113:
4106:
4102:
4096:
4086:
4076:
4070:
4064:
4058:
4051:
4044:
4040:
4034:
4027:
4023:
4020:
3886:
3883:
3765:
3520:is given by
3515:
3511:golden ratio
3483:
3470:
3462:
3454:
3446:
3438:
3431:
3427:
3424:
3280:
3276:
3272:
3268:
3264:
3260:
3210:
3127:
3041:
3020:
3011:
3000:Galois group
2997:
2814:
2810:
2806:
2795:
2788:
2778:
2764:
2754:
2746:
2737:
2734:
2723:
2714:
2705:
2702:
2604:
2443:
2437:
2431:
2428:
2248:
2237:
2226:
2222:
2218:
2215:
2199:
2195:
1637:
1236:
1230:
1226:
1223:
803:
709:
700:
692:
688:
682:
574:
562:group theory
555:
550:
547:
542:
540:
485:
481:
477:
473:
469:
465:
458:
446:
439:
435:
407:
401:Finding the
400:
378:
372:
264:
258:
251:
247:
244:
225:
205:real numbers
170:
53:of the form
46:
40:
9608:Homogeneous
9603:Square-free
9598:Irreducible
9463:Polynomials
9426:AMS Notices
9345:Ragni Piene
8940:= 1.5 x 10
8865:Hill sphere
7708:icosahedron
7704:Felix Klein
7670:. In 1858,
3033:(1 2 3 4 5)
3027:, of order
2244:irreducible
263:produces a
43:mathematics
9679:Categories
9568:Univariate
9196:References
9168:: 275–282.
9121:Klein 1888
7702:. Later,
4049:of degree
3884:where the
3054:, such as
2827:such that
2240:Carl Runge
463:, such as
236:derivative
217:polynomial
9685:Equations
9654:Resultant
9593:Trinomial
9573:Bivariate
9373:cite book
9125:Tóth 2002
8881:≈
8747:∓
8707:±
8609:∓
8534:±
8436:±
8227:π
8208:ω
8107:±
8092:ω
8050:±
8031:±
7575:−
7563:−
7551:−
7464:−
7455:−
7419:ℓ
7368:−
7327:ℓ
7261:−
7200:−
7162:ℓ
7091:−
7085:−
7079:−
7031:−
7005:−
6932:π
6899:∑
6829:−
6759:π
6726:∑
6674:−
6649:−
6633:−
6570:π
6537:∑
6509:−
6468:−
6405:π
6372:∑
6322:−
6306:−
6250:π
6234:
6176:−
6160:−
6052:−
6002:−
5926:−
5917:−
5901:−
5885:−
5847:−
5806:−
5768:−
5759:−
5743:−
5727:−
5517:−
5411:−
5379:−
5341:−
5316:−
5264:−
5220:−
5035:−
5012:−
4920:−
4911:−
4895:−
4879:−
4826:−
4800:−
4730:−
4686:−
4660:−
4616:−
4590:−
4542:−
4516:−
4366:ω
4356:−
4328:ω
4268:−
4146:de Moivre
3979:−
3956:−
3705:−
3696:−
3621:−
3600:−
3574:−
3531:−
3380:−
3320:−
3234:−
3228:∈
3225:β
3219:α
3184:−
3174:β
3159:β
3153:−
3147:−
3142:α
3101:−
3076:−
3070:−
3037:(1 2 4 3)
2869:−
2660:−
2646:±
2557:−
2532:−
2393:ν
2378:ν
2360:ν
2345:μ
2314:ν
2299:ν
2284:μ
2200:In 1888,
2141:−
2087:−
2061:−
1948:−
1890:−
1839:−
1707:−
1665:−
1653:Δ
1607:−
1549:−
1505:−
1479:−
1453:−
1421:−
1395:−
1367:−
1348:−
1323:−
1282:−
1152:−
1101:−
1034:−
990:−
908:−
841:−
517:−
511:−
461:reducible
414:quadratic
9639:Division
9588:Binomial
9583:Monomial
9347:(eds.).
8957:See also
7971:and the
7694:, using
4148:quintic
4074:. These
3481:, where
3434:+ 12 = 0
3283:− 21 = 0
1247:, where
1235:, named
408:Solving
375:radicals
245:Setting
51:function
9303:at the
9062:2369502
8146:is the
7894:to the
7643:Jerrard
6220:Roots:
6109:, with
3509:is the
3506:
3493:√
3488:
3475:√
3459:√
3443:√
2783:
2769:
2759:
2742:
2729:
2710:
1229:− 1024
714:
697:
476:+ 1 = (
442:+ 1 = 0
207:or the
18:Quintic
9476:degree
9357:
9295:
9277:
9223:
9060:
8821:. The
8677:
8665:
8653:
8635:
8414:with:
8192:, and
8181:, and
7726:) and
7289:where
6890:Root:
6717:Root:
6702:
6698:
6528:Root:
6363:Root:
5200:Root:
5188:
5184:
4429:septic
4412:where
3468:, and
3436:. Let
3211:where
3048:nested
2703:where
2429:where
804:where
410:linear
223:five.
221:degree
211:, and
203:, the
171:where
9129:p. 66
9058:JSTOR
8997:(PDF)
8979:Notes
8949:and L
8859:and L
8190:Earth
5929:11200
4109:) = 0
4047:) = 0
4030:) = 0
3002:is a
484:+ 1)(
480:+ 1)(
418:cubic
403:roots
383:cubic
254:) = 0
197:field
49:is a
9465:and
9379:link
9355:ISBN
9293:ISBN
9275:ISBN
9221:ISBN
9185:XLVI
9147:XLVI
8831:and
8814:for
8804:and
8797:for
8154:the
8135:and
7984:SOHO
7982:and
7969:Gaia
7960:and
7934:See
7927:and
6115:= 10
5920:2000
5850:5888
5841:1000
5689:8320
5082:3125
4977:3750
4923:7412
4914:1505
4619:1000
3279:+ 55
3275:− 50
3271:+ 30
3035:and
2823:and
2800:and
2717:+ 3)
2435:and
2016:2250
1951:3750
1922:2000
1842:1600
1694:3125
1638:and
1610:1600
1575:4000
1192:3125
1085:3125
683:the
564:and
488:− 1)
420:and
385:and
191:and
45:, a
9474:By
9264:101
9026:doi
9022:151
8194:Sun
7986:at
7975:at
7910:= 0
7662:= 0
6231:cos
6119:+ 1
6105:th
6080:208
6071:170
6014:800
6005:360
5967:110
5904:600
5825:160
5771:656
5762:150
5680:800
5642:110
5344:400
5335:250
5079:979
5066:125
5063:462
4963:625
4829:300
4593:100
4054:– 1
3430:− 5
3267:− 5
2817:= 0
2796:If
2791:= 0
2726:+ 1
2713:5(4
2251:= 0
2229:= 0
2090:630
2045:108
1973:825
1893:900
1864:144
1809:108
1793:256
1735:560
1668:128
1594:320
1530:224
1482:176
1386:400
1370:240
1126:125
1104:625
1052:125
974:125
261:≠ 0
219:of
41:In
9681::
9424:,
9375:}}
9371:{{
9343:;
9316:+
9314:cx
9312:+
9287:,
9262:,
9258:,
9253:+
9251:ax
9249:+
9239:46
9237:,
9205:,
9183:.
9164:.
9145:.
9093:.
9054:10
9005:36
9003:.
8999:.
8196:.
8174:,
8158:,
8150:,
7912:.
7906:+
7902:−
7748:A
7734:.
7728:11
7710:,
7658:−
7654:+
6952:71
6938:23
6914:13
6864:25
6848:37
6832:28
6779:61
6765:21
6741:11
6677:13
6668:41
6652:17
6636:24
6584:41
6503:21
6471:16
6419:31
6325:21
6309:12
6254:11
6055:20
5989:20
5888:50
5809:40
5746:80
5730:20
5664:60
5602:13
5586:13
5583:10
5541:65
5538:29
5525:13
5480:17
5477:21
5464:17
5461:20
5416:13
5400:85
5319:20
5163:10
5154:30
5138:20
5122:20
5043:25
5040:11
5017:22
4898:20
4882:10
4803:25
4759:15
3491:1+
3486:=
3473:=
3457:=
3452:,
3441:=
3150:10
3073:10
3039:.
3029:20
2970:10
2925:11
2901:10
2813:+
2811:ax
2809:+
2767:=
2762:,
2740:=
2708:=
2657:20
2643:20
2590:0.
2560:11
2235:.
2225:+
2223:ax
2221:+
2115:16
2064:27
1767:16
1710:72
1552:64
1508:80
1456:16
1437:28
1351:16
1298:20
1155:25
1015:15
993:50
941:25
911:15
892:25
695:−
691:=
472:−
468:−
451:.
444:.
438:−
416:,
412:,
393:.
357:0.
242:.
187:,
183:,
179:,
175:,
9455:e
9448:t
9441:v
9381:)
9321:.
9318:d
9310:x
9255:b
9247:x
9229:.
9207:2
9166:I
9131:)
9119:(
9097:.
9028::
8951:2
8947:1
8942:m
8938:r
8917:3
8909:S
8905:M
8901:3
8895:E
8891:M
8884:R
8878:r
8861:2
8857:1
8853:S
8850:M
8846:E
8843:M
8836:1
8833:L
8829:2
8826:L
8819:1
8816:L
8811:m
8807:r
8802:2
8799:L
8794:m
8790:r
8783:.
8765:5
8761:R
8755:E
8751:M
8744:=
8741:f
8733:,
8728:4
8724:R
8720:2
8715:E
8711:M
8704:=
8701:e
8693:,
8690:)
8685:2
8681:L
8674:r
8671:o
8668:f
8662:0
8659:=
8656:d
8650:s
8647:u
8644:h
8641:t
8638:(
8630:3
8626:R
8622:)
8617:E
8613:M
8604:E
8600:M
8596:(
8593:+
8590:=
8587:d
8579:,
8574:2
8570:R
8566:3
8563:)
8558:E
8554:M
8550:+
8545:S
8541:M
8537:(
8531:=
8528:c
8520:,
8517:R
8514:3
8511:)
8506:E
8502:M
8498:+
8493:S
8489:M
8485:(
8482:+
8479:=
8476:b
8468:,
8465:)
8460:E
8456:M
8452:+
8447:S
8443:M
8439:(
8433:=
8430:a
8399:0
8396:=
8393:f
8390:+
8387:r
8384:e
8381:+
8376:2
8372:r
8368:d
8365:+
8360:3
8356:r
8352:c
8349:+
8344:4
8340:r
8336:b
8333:+
8328:5
8324:r
8320:a
8293:3
8289:R
8284:)
8279:E
8275:M
8271:+
8266:S
8262:M
8258:(
8255:G
8249:=
8242:2
8238:P
8231:2
8223:4
8217:=
8212:2
8185:S
8183:M
8178:E
8176:M
8172:m
8164:R
8160:r
8152:ω
8144:G
8140:1
8137:L
8133:2
8130:L
8113:)
8110:r
8104:R
8101:(
8096:2
8088:m
8085:=
8078:2
8074:r
8067:E
8063:M
8059:m
8056:G
8042:2
8038:)
8034:r
8028:R
8025:(
8018:S
8014:M
8010:m
8007:G
7991:1
7988:L
7980:2
7977:L
7965:1
7962:L
7958:2
7955:L
7908:t
7904:x
7900:x
7878:0
7875:=
7870:0
7866:a
7862:+
7859:x
7854:1
7850:a
7846:+
7841:2
7837:x
7831:2
7827:a
7823:+
7818:3
7814:x
7808:3
7804:a
7800:+
7795:4
7791:x
7785:4
7781:a
7777:+
7772:5
7768:x
7722:(
7720:7
7667:a
7660:a
7656:t
7652:t
7598:.
7593:]
7588:m
7583:2
7579:a
7572:)
7569:m
7566:4
7560:a
7557:(
7554:p
7548:)
7545:m
7542:4
7539:+
7536:a
7533:(
7530:b
7525:[
7518:2
7515:1
7509:=
7506:c
7481:,
7475:2
7471:m
7467:2
7461:p
7458:5
7452:)
7447:2
7443:a
7439:+
7434:2
7430:m
7426:4
7423:(
7416:=
7413:b
7388:,
7383:]
7376:2
7372:m
7365:)
7360:2
7356:a
7352:+
7347:2
7343:m
7339:4
7336:(
7331:2
7321:[
7314:4
7311:1
7305:=
7302:p
7274:0
7271:=
7268:c
7264:p
7257:)
7253:x
7249:b
7246:+
7241:2
7237:x
7232:a
7229:+
7224:3
7220:x
7215:2
7211:(
7207:p
7203:5
7195:5
7191:x
7168:m
7165:,
7159:,
7156:a
7133:0
7130:=
7127:a
7124:+
7121:x
7117:b
7114:+
7109:2
7105:x
7100:)
7097:b
7094:2
7088:1
7082:a
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7070:a
7066:(
7063:+
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7046:3
7043:+
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7037:+
7034:a
7028:(
7025:+
7020:4
7016:x
7011:)
7008:3
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6999:(
6996:+
6991:5
6987:x
6946:k
6942:)
6935:(
6929:i
6926:2
6920:e
6909:0
6906:=
6903:k
6873:1
6870:+
6867:x
6861:+
6856:2
6852:x
6845:+
6840:3
6836:x
6824:4
6820:x
6816:+
6811:5
6807:x
6773:k
6769:)
6762:(
6756:i
6753:2
6747:e
6736:0
6733:=
6730:k
6671:x
6665:+
6660:2
6656:x
6644:3
6640:x
6628:4
6624:x
6620:+
6615:5
6611:x
6578:k
6574:3
6567:i
6564:2
6558:e
6552:7
6547:0
6544:=
6541:k
6512:9
6506:x
6500:+
6495:2
6491:x
6487:5
6484:+
6479:3
6475:x
6463:4
6459:x
6455:+
6450:5
6446:x
6413:k
6409:6
6402:i
6399:2
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6387:5
6382:0
6379:=
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6338:+
6333:2
6329:x
6317:3
6313:x
6301:4
6297:x
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5981:3
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5923:x
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5880:5
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5844:x
5838:+
5833:2
5829:x
5822:+
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5813:x
5801:5
5797:x
5765:x
5754:2
5750:x
5738:3
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5722:5
5718:x
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5668:x
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5630:x
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5469:x
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5210:2
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4949:x
4917:x
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4886:x
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4837:5
4833:s
4821:2
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4725:5
4721:x
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4623:s
4611:2
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4537:2
4533:x
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4507:x
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4421:ω
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4397:,
4388:5
4382:i
4378:y
4370:k
4361:a
4350:5
4344:i
4340:y
4332:k
4324:=
4319:k
4315:x
4288:,
4284:0
4281:=
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4272:a
4265:y
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4259:+
4254:2
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4220:,
4216:0
4213:=
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4204:x
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4183:3
4179:x
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4120:p
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4105:(
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4097:Q
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4065:p
4059:P
4052:p
4045:y
4043:(
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4035:p
4028:x
4026:(
4024:P
4006:.
4002:0
3999:=
3992:5
3988:5
3984:1
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3969:3
3965:5
3961:8
3951:2
3947:y
3941:5
3938:4
3933:+
3928:3
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3912:4
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3869:,
3862:5
3856:4
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3846:+
3840:5
3834:3
3830:y
3824:+
3818:5
3812:2
3808:y
3802:+
3796:5
3790:1
3786:y
3780:=
3777:x
3751:,
3744:5
3739:)
3736:c
3733:+
3730:b
3727:(
3722:2
3718:)
3714:c
3711:+
3708:a
3702:(
3690:5
3683:2
3679:)
3675:c
3672:+
3669:b
3666:(
3663:)
3660:c
3657:+
3654:a
3651:(
3645:+
3639:5
3632:2
3628:)
3624:c
3618:b
3615:(
3612:)
3609:c
3606:+
3603:a
3597:(
3591:+
3585:5
3580:)
3577:c
3571:b
3568:(
3563:2
3559:)
3555:c
3552:+
3549:a
3546:(
3540:=
3537:x
3534:c
3516:x
3503:2
3500:/
3495:5
3484:φ
3477:5
3471:c
3463:φ
3461:2
3455:b
3447:φ
3445:2
3439:a
3432:x
3428:x
3410:.
3405:4
3400:)
3394:5
3390:2
3385:(
3375:3
3370:)
3364:5
3360:2
3355:(
3350:+
3345:2
3340:)
3334:5
3330:2
3325:(
3314:5
3310:2
3305:+
3302:1
3299:=
3296:x
3281:x
3277:x
3273:x
3269:x
3265:x
3246:}
3243:1
3240:,
3237:1
3231:{
3222:,
3196:,
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3091:+
3084:5
3079:2
3024:5
3021:F
3015:5
3012:S
2978:.
2966:l
2962:+
2957:2
2953:m
2947:)
2944:m
2941:2
2938:+
2933:5
2929:l
2922:(
2919:4
2913:=
2910:b
2897:l
2893:+
2888:2
2884:m
2878:)
2875:m
2872:4
2864:5
2860:l
2856:3
2853:(
2850:l
2847:5
2841:=
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2825:m
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2807:x
2802:b
2798:a
2789:a
2779:l
2775:/
2772:1
2765:e
2755:l
2751:/
2747:m
2745:−
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2724:ν
2720:/
2715:ν
2706:a
2687:)
2681:)
2678:a
2675:+
2672:5
2669:(
2666:)
2663:a
2654:(
2649:2
2640:+
2637:a
2633:(
2627:5
2624:4
2619:=
2616:b
2587:=
2581:1
2578:+
2573:2
2569:c
2563:)
2554:c
2551:2
2548:(
2543:5
2539:e
2535:4
2526:+
2523:x
2517:1
2514:+
2509:2
2505:c
2499:)
2496:3
2493:+
2490:c
2487:4
2484:(
2479:4
2475:e
2471:5
2465:+
2460:5
2456:x
2438:ν
2432:μ
2414:0
2411:=
2405:1
2402:+
2397:2
2387:)
2384:3
2381:+
2375:4
2372:(
2369:)
2366:1
2363:+
2357:2
2354:(
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2341:4
2335:+
2332:x
2326:1
2323:+
2318:2
2308:)
2305:3
2302:+
2296:4
2293:(
2288:4
2280:5
2274:+
2269:5
2265:x
2249:a
2227:b
2219:x
2177:.
2172:2
2168:r
2162:2
2158:q
2152:3
2148:p
2144:4
2138:s
2133:3
2129:q
2123:3
2119:p
2112:+
2109:s
2106:r
2101:3
2097:q
2093:p
2082:2
2078:r
2072:4
2068:q
2058:s
2053:5
2049:q
2042:+
2037:2
2033:s
2029:r
2024:2
2020:q
2013:+
2001:2
1997:s
1991:2
1987:q
1981:2
1977:p
1970:+
1965:3
1961:s
1957:q
1954:p
1943:2
1939:s
1933:2
1929:r
1925:p
1919:+
1914:2
1910:s
1906:r
1901:3
1897:p
1885:3
1881:r
1875:2
1871:q
1867:p
1861:+
1858:s
1853:3
1849:r
1845:q
1827:2
1823:s
1817:5
1813:p
1806:+
1801:5
1797:r
1790:+
1785:3
1781:r
1775:4
1771:p
1764:+
1761:s
1756:2
1752:r
1748:q
1743:2
1739:p
1732:+
1729:s
1726:r
1723:q
1718:4
1714:p
1702:4
1698:s
1691:+
1686:4
1682:r
1676:2
1672:p
1656:=
1619:q
1616:s
1613:r
1602:3
1598:r
1591:+
1586:2
1582:s
1578:p
1572:+
1560:4
1556:q
1544:2
1540:q
1536:r
1533:p
1527:+
1524:q
1521:s
1516:2
1512:p
1500:2
1496:r
1490:2
1486:p
1474:2
1470:q
1464:3
1460:p
1450:r
1445:4
1441:p
1434:+
1429:6
1425:p
1411:)
1406:4
1402:p
1398:3
1392:q
1389:s
1383:+
1378:2
1374:r
1362:2
1358:q
1354:p
1345:r
1340:2
1336:p
1332:8
1329:(
1326:z
1320:)
1315:2
1311:p
1307:3
1304:+
1301:r
1295:(
1290:2
1286:z
1277:3
1273:z
1265:=
1262:P
1245:z
1233:Δ
1231:z
1227:P
1200:5
1196:a
1185:5
1181:b
1177:4
1174:+
1171:c
1166:3
1162:b
1158:a
1149:d
1144:2
1140:b
1134:2
1130:a
1123:+
1120:e
1117:b
1112:3
1108:a
1098:f
1093:4
1089:a
1079:=
1072:s
1060:4
1056:a
1045:4
1041:b
1037:3
1031:c
1026:2
1022:b
1018:a
1012:+
1009:d
1006:b
1001:2
997:a
987:e
982:3
978:a
968:=
961:r
949:3
945:a
934:3
930:b
926:4
923:+
920:c
917:b
914:a
905:d
900:2
896:a
886:=
879:q
867:2
863:a
859:5
852:2
848:b
844:2
838:c
835:a
832:5
826:=
819:p
800:,
788:0
785:=
782:s
779:+
776:y
773:r
770:+
765:2
761:y
757:q
754:+
749:3
745:y
741:p
738:+
733:5
729:y
710:a
708:5
705:/
701:b
693:y
689:x
668:,
665:0
662:=
659:f
656:+
653:x
650:e
647:+
642:2
638:x
634:d
631:+
626:3
622:x
618:c
615:+
610:4
606:x
602:b
599:+
594:5
590:x
586:a
543:r
526:0
523:=
520:r
514:x
506:5
502:x
486:x
482:x
478:x
474:x
470:x
466:x
440:x
436:x
379:n
377:(
354:=
351:f
348:+
345:x
342:e
339:+
334:2
330:x
326:d
323:+
318:3
314:x
310:c
307:+
302:4
298:x
294:b
291:+
286:5
282:x
278:a
259:a
252:x
250:(
248:g
213:a
193:f
189:e
185:d
181:c
177:b
173:a
155:,
152:f
149:+
146:x
143:e
140:+
135:2
131:x
127:d
124:+
119:3
115:x
111:c
108:+
103:4
99:x
95:b
92:+
87:5
83:x
79:a
76:=
73:)
70:x
67:(
64:g
20:)
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