2180:
20:
1208:
1633:
8770:
1622:
799:
2175:{\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}}}
8409:
1242:
3250:
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
1203:{\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}}}
2185:
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.
3750:
3409:
2413:
537:
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
5083:
413:
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
2977:
8112:
8765:{\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}}}
4005:
2589:
1617:{\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}}}
3195:
4396:
8296:
7621:
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).
7132:
3868:
2687:
7878:
5282:
3112:
7273:
6954:
6781:
5420:
6586:
6421:
8919:
5481:
155:
6259:
5603:
5542:
357:
7480:
4558:
8414:
5691:
1638:
1247:
804:
667:
8398:
6676:
6016:
5928:
6872:
5849:
6511:
4922:
422:, 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
6346:
3515:
6203:
5770:
4628:
5162:
4976:
4219:
4838:
3280:
3245:
2248:
787:
7597:
7387:
4768:
4698:
436:
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
6079:
5343:
7703:, 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
4287:
4986:
4478:
2822:
525:
7167:
6701:
5187:
7988:
2808:
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
3891:
2439:
4124:-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.
3123:
1213:
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
9329:
4298:
8191:
3039:. 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
7956:
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
4133:
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
6970:
3761:
2721:. Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second.
3117:
In fact, all four primitive fifth roots of unity may be obtained by changing the signs of the square roots appropriately; namely, the expression
2600:
9367:
7751:
5192:
3049:
9337:
7174:
9401:
7615:, there are solvable quintics which have five real roots all of whose solutions in radicals involve roots of complex numbers. This is
6882:
6709:
5353:
418:(that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the
9289:
6520:
6355:
448:
Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is
8862:
5430:
48:
9442:
9168:
Kronecker, Leopold (1858). "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite".
6212:
5552:
5491:
378:
were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the
437:
8811:
262:
7397:
4490:
7704:
5613:
570:
8304:
6594:
5938:
5859:
4428:
There are infinitely many solvable quintics in Bring–Jerrard form which have been parameterized in a preceding section.
3745:{\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)}}\,,}
9622:
7972:
6790:
5780:
6429:
4853:
9347:
9285:
9267:
9222:
Leopold
Kronecker, "Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite",
9213:
6267:
6121:
5701:
4564:
3404:{\displaystyle x=1+{\sqrt{2}}-\left({\sqrt{2}}\right)^{2}+\left({\sqrt{2}}\right)^{3}-\left({\sqrt{2}}\right)^{4}.}
2408:{\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}
5093:
4932:
4143:
561:
found a general criterion for determining whether any given quintic is solvable. This criterion is the following.
4774:
3414:
An example of a more complicated (although small enough to be written here) solution is the unique real root of
3203:
712:
7490:
7286:
4704:
7982:) provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun:
7720:
4634:
9683:
7884:
7939:
of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic.
6026:
5290:
9617:
9601:
24:
9149:
Brioschi, Francesco (1858). "Sul Metodo di
Kronecker per la Risoluzione delle Equazioni di Quinto Grado".
3007:
of all permutations of a five element set, which is solvable if and only if it is a subgroup of the group
9637:
9320:
Ehrenfried Walter von
Tschirnhaus, "A method for removing all intermediate terms from a given equation,"
9313:
Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of
Tschirnhaus, Bring and Jerrard,"
4233:
5078:{\displaystyle x^{5}-{\frac {22}{5}}x^{3}-{\frac {11}{25}}x^{2}+{\frac {462}{125}}x+{\frac {979}{3125}}}
9482:
7961:
7738:
673:
2972:{\displaystyle a={\frac {5l(3l^{5}-4m)}{m^{2}+l^{10}}}\qquad b={\frac {4(11l^{5}+2m)}{m^{2}+l^{10}}}.}
9435:
9407:
7668:
4434:
2594:
The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression
2193:
described how to solve a solvable quintic equation, without providing an explicit formula; in 2004,
9627:
419:
379:
9566:
4413:
485:
9353:
8107:{\displaystyle {\frac {GmM_{S}}{(R\pm r)^{2}}}\pm {\frac {GmM_{E}}{r^{2}}}=m\omega ^{2}(R\pm r)}
9678:
9596:
9591:
9586:
9464:
9270:. Discusses Galois Theory in general including a proof of insolvability of the general quintic.
8136:
7676:
7140:
6682:
5168:
2232:
2221:
449:
209:
39:
4027:
with rational coefficients is solvable in radicals, then one can define an auxiliary equation
9576:
9556:
9117:
4070:
706:, which depresses the quintic (that is, removes the term of degree four), gives the equation
545:
To characterize solvable quintics, and more generally solvable polynomials of higher degree,
542:
is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals.
4000:{\displaystyle y^{4}+4y^{3}+{\frac {4}{5}}y^{2}-{\frac {8}{5^{3}}}y-{\frac {1}{5^{5}}}=0\,.}
9642:
9561:
9428:
4224:
where the auxiliary equation has two zero roots and reduces, by factoring them out, to the
3032:
2584:{\displaystyle x^{5}+{\frac {5e^{4}(4c+3)}{c^{2}+1}}x+{\frac {-4e^{5}(2c-11)}{c^{2}+1}}=0.}
2235:
quintic with rational coefficients in Bring–Jerrard form is solvable if and only if either
415:
6964:
There are also two parameterized families of solvable quintics: The Kondo–Brumer quintic,
2227:
During the second half of the 19th century, John Stuart
Glashan, George Paxton Young, and
8:
9455:
8961:
7617:
4431:
Up to the scaling of the variable, there are exactly five solvable quintics of the shape
3190:{\displaystyle {\frac {\alpha {\sqrt {-10-2\beta {\sqrt {5}}}}+\beta {\sqrt {5}}-1}{4}},}
2190:
185:
9395:
9296: (archived 31 March 2010)) gives a description of the solution of solvable quintics
9273:
9097:
9673:
9632:
9499:
9494:
9420:
9361:
9083:
9046:
9006:
A. Cayley, "On a new auxiliary equation in the theory of equation of the fifth order",
8983:"Equations of the Bring–Jerrard Form, the Golden Section, and Square Fibonacci Numbers"
7913:
7688:
4391:{\displaystyle x_{k}=\omega ^{k}{\sqrt{y_{i}}}-{\frac {a}{\omega ^{k}{\sqrt{y_{i}}}}},}
4225:
4080:
402:
391:
9075:
8291:{\displaystyle \omega ^{2}={\frac {4\pi ^{2}}{P^{2}}}={\frac {G(M_{S}+M_{E})}{R^{3}}}}
6090:
An infinite sequence of solvable quintics may be constructed, whose roots are sums of
546:
9477:
9389:
9343:
9281:
9263:
9209:
7917:
7909:
7680:
2775:
in the
Spearman-Williams parameterization allows one to not exclude the special case
220:
9523:
9518:
9038:
George Paxton Young, "Solvable
Quintic Equations with Commensurable Coefficients",
9014:
8144:
7936:
7742:
3883:
410:
375:
228:
215:
Because they have an odd degree, normal quintic functions appear similar to normal
9652:
9540:
9535:
9487:
9472:
9293:
8956:
8951:
8156:
7905:
7712:
7660:
7127:{\displaystyle x^{5}+(a-3)\,x^{4}+(-a+b+3)\,x^{3}+(a^{2}-a-1-2b)\,x^{2}+b\,x+a=0}
4417:
2996:
398:
189:
3863:{\displaystyle x={\sqrt{y_{1}}}+{\sqrt{y_{2}}}+{\sqrt{y_{3}}}+{\sqrt{y_{4}}}\,,}
534:
is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible.
9511:
9506:
9233:
Blair
Spearman and Kenneth S. Williams, "Characterization of solvable quintics
9203:
9130:
Hermite, Charles (1858). "Sur la résolution de l'équation du cinquième degré".
8982:
7672:
7664:
7631:
7612:
6095:
3036:
2992:
406:
371:
216:
197:
7904:
The roots of this equation cannot be expressed by radicals. However, in 1858,
2205:
There are several parametric representations of solvable quintics of the form
9667:
9205:
Lectures on the
Icosahedron and the Solution of Equations of the Fifth Degree
7924:
7732:
7700:
3040:
2995:. In the case of irreducible quintics, the Galois group is a subgroup of the
2194:
558:
554:
7663:
showed that the Bring radical could be characterized in terms of the Jacobi
9647:
9018:
7684:
7635:
3499:
2988:
2682:{\displaystyle b={\frac {4}{5}}\left(a+20\pm 2{\sqrt {(20-a)(5+a)}}\right)}
550:
9029:
This formulation of Cayley's result is extracted from Lazard (2004) paper.
3031:
If the quintic is solvable, one of the solutions may be represented by an
9414:
9333:
8853:
7696:
7692:
394:(zeros) of a given polynomial has been a prominent mathematical problem.
193:
31:
9451:
9050:
7957:
7873:{\displaystyle x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\,}
5277:{\displaystyle {\sqrt{2}}-{\sqrt{2}}^{2}+{\sqrt{2}}^{3}-{\sqrt{2}}^{4}}
2228:
530:
has solutions in radicals if and only if it has an integer solution or
224:
205:
9190:
Charles
Hermite, "Sur la résolution de l'équation du cinquème degré",
3107:{\displaystyle {\frac {{\sqrt {-10-2{\sqrt {5}}}}+{\sqrt {5}}-1}{4}}.}
9581:
7671:, using an approach similar to the more familiar approach of solving
4134:
3251:
form of the first solution may be rather simple, as for the equation
2433:
In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,
370:
th roots) was a major problem in algebra from the 16th century, when
19:
9116:, Section 1.6, Additional Topic: Klein's Theory of the Icosahedron,
7268:{\displaystyle x^{5}-5\,p\left(2\,x^{3}+a\,x^{2}+b\,x\right)-p\,c=0}
4843:
Paxton Young (1888) gave a number of examples of solvable quintics:
9571:
363:
23:
Graph of a polynomial of degree 5, with 3 real zeros (roots) and 4
9402:
A method for removing all intermediate terms from a given equation
9398:– a method for solving solvable quintics due to David S. Dummit.
7908:
published the first known solution of this equation in terms of
4106:-th roots may not be computed independently (this would provide
9377:
Finite Möbius groups, minimal immersions of spheres, and moduli
7687:, developed a simpler way of deriving Hermite's result, as had
4292:
such that the five roots of the de Moivre quintic are given by
3035:
involving a fifth root and at most two square roots, generally
204:
is nonzero. In other words, a quintic function is defined by a
6949:{\displaystyle \sum _{k=0}^{13}e^{\frac {2i\pi (23)^{k}}{71}}}
6776:{\displaystyle \sum _{k=0}^{11}e^{\frac {2i\pi (21)^{k}}{61}}}
9151:
Atti Dell'i. R. Istituto Lombardo di Scienze, Lettere ed Arti
8178:
5415:{\displaystyle x^{5}-5x^{3}+{\frac {85}{8}}x-{\frac {13}{2}}}
3247:, yields the four distinct primitive fifth roots of unity.
9404:- a recent English translation of Tschirnhaus' 1683 paper.
9328:
Lazard, Daniel (2004). "Solving quintics in radicals". In
6581:{\displaystyle \sum _{k=0}^{7}e^{\frac {2i\pi 3^{k}}{41}}}
6416:{\displaystyle \sum _{k=0}^{5}e^{\frac {2i\pi 6^{k}}{31}}}
4416:. This can be easily generalized to construct a solvable
4045:, also with rational coefficients, such that each root of
9208:. Translated by Morrice, George Gavin. Trübner & Co.
9008:
Philosophical Transactions of the Royal Society of London
8844:), then the quintic equation can be greatly reduced and L
8182:
7695:
came up with a method that relates the symmetries of the
9450:
8914:{\displaystyle r\approx R{\sqrt{\frac {M_{E}}{3M_{S}}}}}
7638:(also known as Bring radicals), the unique real root of
5476:{\displaystyle x^{5}+{\frac {20}{17}}x+{\frac {21}{17}}}
150:{\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,}
6254:{\displaystyle 2\cos \left({\frac {2k\pi }{11}}\right)}
5598:{\displaystyle x^{5}+{\frac {10}{13}}x+{\frac {3}{13}}}
5537:{\displaystyle x^{5}-{\frac {4}{13}}x+{\frac {29}{65}}}
8837:) is much smaller than the mass of the larger object (
7927:
for details on these solutions and some related ones.
7501:
7297:
4118:). Thus a correct solution needs to express all these
385:
9278:
Galois theory for beginners: A historical perspective
8865:
8412:
8307:
8194:
7991:
7930:
7754:
7493:
7400:
7289:
7177:
7143:
6973:
6885:
6793:
6712:
6685:
6597:
6523:
6432:
6358:
6270:
6215:
6124:
6029:
5941:
5862:
5783:
5704:
5616:
5555:
5494:
5433:
5356:
5293:
5195:
5171:
5096:
4989:
4935:
4856:
4777:
4707:
4637:
4567:
4493:
4437:
4301:
4236:
4146:
3894:
3764:
3518:
3283:
3206:
3126:
3052:
2987:
A polynomial equation is solvable by radicals if its
2825:
2603:
2442:
2251:
1636:
1245:
802:
715:
573:
488:
352:{\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,}
265:
219:
when graphed, except they may possess one additional
51:
7745:, reduces the general quintic equation of the form
7475:{\displaystyle b=\ell \,(4m^{2}+a^{2})-5p-2m^{2}\;,}
4553:{\displaystyle x^{5}-2s^{3}x^{2}-{\frac {s^{5}}{5}}}
4408:
is any root of the auxiliary quadratic equation and
5686:{\displaystyle x^{5}+110(5x^{3}+60x^{2}+800x+8320)}
662:{\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0,}
8913:
8764:
8393:{\displaystyle ar^{5}+br^{4}+cr^{3}+dr^{2}+er+f=0}
8392:
8290:
8106:
7872:
7634:demonstrated that quintics can be solved by using
7591:
7474:
7381:
7267:
7161:
7126:
6948:
6866:
6775:
6695:
6671:{\displaystyle x^{5}+x^{4}-24x^{3}-17x^{2}+41x-13}
6670:
6580:
6505:
6415:
6340:
6253:
6197:
6073:
6011:{\displaystyle x^{5}+110(5x^{3}+20x^{2}-360x+800)}
6010:
5923:{\displaystyle x^{5}-50x^{3}-600x^{2}-2000x-11200}
5922:
5843:
5764:
5685:
5597:
5536:
5475:
5414:
5337:
5276:
5181:
5156:
5077:
4970:
4916:
4832:
4762:
4692:
4622:
4552:
4472:
4390:
4281:
4213:
3999:
3862:
3744:
3403:
3239:
3189:
3106:
2971:
2681:
2583:
2407:
2174:
1616:
1202:
781:
661:
519:
351:
149:
9410:The Quintic, the Icosahedron, and Elliptic Curves
9317:, Vol. 37, No. 3, September 2003, pp. 90–94.
6867:{\displaystyle x^{5}+x^{4}-28x^{3}+37x^{2}+25x+1}
5844:{\displaystyle x^{5}-40x^{3}+160x^{2}+1000x-5888}
2200:
9665:
8827:are usually given as 1.5 million km from Earth.
6506:{\displaystyle x^{5}+x^{4}-16x^{3}+5x^{2}+21x-9}
4917:{\displaystyle x^{5}-10x^{3}-20x^{2}-1505x-7412}
9392:– more details on methods for solving Quintics.
6341:{\displaystyle x^{5}+x^{4}-12x^{3}-21x^{2}+x+5}
9473:Zero polynomial (degree undefined or −1 or −∞)
7726:
7721:Icosahedral symmetry § Related geometries
6198:{\displaystyle x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1}
5765:{\displaystyle x^{5}-20x^{3}-80x^{2}-150x-656}
4420:and other odd degrees, not necessarily prime.
2982:
9436:
9290:The solution of equations of the fifth degree
8980:
8298:and rearranging all terms yields the quintic
4623:{\displaystyle x^{5}-100s^{3}x^{2}-1000s^{5}}
5157:{\displaystyle x^{5}+20x^{3}+20x^{2}+30x+10}
4971:{\displaystyle x^{5}+{\frac {625}{4}}x+3750}
4214:{\displaystyle x^{5}+5ax^{3}+5a^{2}x+b=0\,,}
3234:
3219:
9324:, Vol. 37, No. 1, March 2003, pp. 1–3.
7137:and the family depending on the parameters
4833:{\displaystyle x^{5}-25s^{3}x^{2}-300s^{5}}
9443:
9429:
9366:: CS1 maint: location missing publisher (
7585:
7468:
7375:
4423:
3240:{\displaystyle \alpha ,\beta \in \{-1,1\}}
782:{\displaystyle y^{5}+py^{3}+qy^{2}+ry+s=0}
9224:Comptes Rendus de l'Académie des Sciences
9170:Comptes Rendus de l'Académie des Sciences
9167:
9132:Comptes Rendus de l'Académie des Sciences
7869:
7592:{\displaystyle c={\tfrac {1}{2}}\left\;.}
7579:
7517:
7410:
7382:{\displaystyle p={\tfrac {1}{4}}\left\;,}
7369:
7313:
7255:
7240:
7223:
7206:
7194:
7108:
7091:
7040:
7002:
4763:{\displaystyle x^{5}-5s^{3}x^{2}+15s^{5}}
4275:
4207:
4127:It is possible that some of the roots of
3993:
3856:
3738:
348:
146:
9148:
8177:are the respective masses of satellite,
8155:the distance Sun to Earth (that is, the
8151:the distance of the satellite to Earth,
4693:{\displaystyle x^{5}-5s^{3}x^{2}-3s^{5}}
549:developed techniques which gave rise to
18:
9280:, American Mathematical Society, 2006.
9129:
8852:are at approximately the radius of the
7719:, as studied by Klein and discussed in
3020:, generated by the cyclic permutations
9666:
9327:
9123:
9061:
7604:
6074:{\displaystyle x^{5}-20x^{3}+170x+208}
5338:{\displaystyle x^{5}-20x^{3}+250x-400}
3274:, for which the only real solution is
479:. For example, it has been shown that
438:root-finding algorithm for polynomials
362:Solving quintic equations in terms of
223:and one additional local minimum. The
9424:
9262:2nd Edition, Chapman and Hall, 1989.
9201:
9109:
7741:, which may be computed by solving a
9374:
9113:
9112:); a modern exposition is given in (
7707:. Similar phenomena occur in degree
7705:generalized hypergeometric functions
443:
8830:If the mass of the smaller object (
4282:{\displaystyle y^{2}+by-a^{5}=0\,,}
4089:and its roots can be used to solve
2793:are rational numbers, the equation
386:Finding roots of a quintic equation
13:
9073:
8776:Solving these two quintics yields
7931:Application to celestial mechanics
7625:
1641:
14:
9695:
9383:
7942:More precisely, the locations of
7935:Solving for the locations of the
2231:gave such a parameterization: an
8981:Elia, M.; Filipponi, P. (1998).
7920:came upon equivalent solutions.
2197:wrote out a three-page formula.
9339:The Legacy of Niels Henrik Abel
9161:
9142:
9080:with interesting Galois groups"
9040:American Journal of Mathematics
6689:
5175:
4010:More generally, if an equation
2897:
2782:, giving the following result:
16:Polynomial function of degree 5
9198::5–21, Gauthier-Villars, 1908.
9102:
9090:
9067:
9055:
9032:
9023:
9000:
8974:
8678:
8626:
8610:
8584:
8551:
8525:
8499:
8473:
8453:
8427:
8272:
8246:
8101:
8089:
8026:
8013:
7560:
7545:
7536:
7521:
7440:
7411:
7353:
7324:
7088:
7054:
7037:
7016:
6999:
6987:
6930:
6923:
6757:
6750:
6005:
5958:
5680:
5633:
4473:{\displaystyle x^{5}+ax^{2}+b}
3727:
3715:
3706:
3690:
3667:
3654:
3651:
3639:
3616:
3603:
3600:
3585:
3568:
3556:
3547:
3534:
3502:. Then the only real solution
2935:
2910:
2866:
2841:
2669:
2657:
2654:
2642:
2551:
2536:
2487:
2472:
2375:
2360:
2357:
2342:
2296:
2281:
2201:Quintics in Bring–Jerrard form
1399:
1317:
1308:
1283:
61:
55:
1:
9633:Horner's method of evaluation
9249:American Mathematical Monthly
9184:
4069:-th roots were introduced by
557:. Applying these techniques,
9390:Mathworld - Quintic Equation
9342:. Berlin. pp. 207–225.
7679:. At around the same time,
4414:primitive 5th roots of unity
7:
9638:Polynomial identity testing
8945:
8812:Sun–Earth Lagrangian points
7727:Solving with Bring radicals
3041:primitive 5th root of unity
2983:Roots of a solvable quintic
520:{\displaystyle x^{5}-x-r=0}
227:of a quintic function is a
10:
9700:
8117:The ± sign corresponds to
7962:James Webb Space Telescope
7912:. At around the same time
7739:Tschirnhaus transformation
7730:
7669:elliptic modular functions
4057:-th roots of the roots of
3882:are the four roots of the
674:Tschirnhaus transformation
9610:
9549:
9462:
9396:Solving Solvable Quintics
9192:Œuvres de Charles Hermite
9098:Solving Solvable Quintics
8942:in the Sun-Earth system.
8188:Using Kepler's Third Law
7885:Bring–Jerrard normal form
7162:{\displaystyle a,\ell ,m}
6696:{\displaystyle ~\qquad ~}
5182:{\displaystyle ~\qquad ~}
1232:, has a rational root in
8967:
4073:, and their products by
9623:Greatest common divisor
8990:The Fibonacci Quarterly
8159:of Earth's orbit), and
7677:trigonometric functions
4424:Other solvable quintics
9495:Quadratic function (2)
9019:10.1098/rstl.1861.0014
8915:
8766:
8394:
8292:
8137:gravitational constant
8108:
7874:
7593:
7476:
7383:
7269:
7163:
7128:
6950:
6906:
6868:
6777:
6733:
6697:
6672:
6582:
6544:
6507:
6417:
6379:
6342:
6255:
6199:
6111:being a prime number:
6075:
6012:
5924:
5845:
5766:
5687:
5599:
5538:
5477:
5416:
5339:
5278:
5183:
5158:
5079:
4972:
4918:
4834:
4764:
4694:
4624:
4554:
4484:is a scaling factor):
4474:
4392:
4283:
4215:
4001:
3864:
3746:
3405:
3241:
3191:
3108:
2973:
2683:
2585:
2409:
2176:
1618:
1204:
783:
663:
521:
353:
151:
27:
9478:Constant function (0)
9202:Klein, Felix (1888).
8916:
8767:
8395:
8293:
8109:
7875:
7667:and their associated
7594:
7477:
7384:
7270:
7164:
7129:
6951:
6886:
6869:
6778:
6713:
6698:
6673:
6583:
6524:
6508:
6418:
6359:
6343:
6256:
6200:
6076:
6013:
5925:
5846:
5767:
5688:
5600:
5539:
5478:
5417:
5340:
5279:
5184:
5159:
5080:
4973:
4919:
4835:
4765:
4695:
4625:
4555:
4475:
4393:
4284:
4216:
4083:. The computation of
4071:Joseph-Louis Lagrange
4002:
3865:
3755:or, equivalently, by
3747:
3406:
3242:
3192:
3109:
2974:
2684:
2586:
2410:
2242:or it may be written
2177:
1619:
1205:
784:
664:
522:
354:
152:
22:
9684:Polynomial functions
9611:Tools and algorithms
9531:Quintic function (5)
9519:Quartic function (4)
9456:polynomial functions
9375:Tóth, Gábor (2002),
8863:
8410:
8305:
8192:
7989:
7752:
7491:
7398:
7287:
7175:
7141:
6971:
6883:
6791:
6710:
6683:
6595:
6521:
6430:
6356:
6268:
6213:
6122:
6027:
5939:
5860:
5781:
5702:
5614:
5553:
5492:
5431:
5354:
5291:
5193:
5169:
5094:
4987:
4933:
4854:
4775:
4705:
4635:
4565:
4491:
4435:
4299:
4234:
4144:
4079:are commonly called
3892:
3762:
3516:
3281:
3204:
3124:
3050:
3033:algebraic expression
2823:
2601:
2440:
2249:
1634:
1243:
800:
713:
571:
486:
420:Abel–Ruffini theorem
416:algebraic expression
380:Abel–Ruffini theorem
263:
49:
9541:Septic equation (7)
9536:Sextic equation (6)
9483:Linear function (1)
9356:on January 6, 2005.
9330:Olav Arnfinn Laudal
9322:ACM SIGSAM Bulletin
9315:ACM SIGSAM Bulletin
8962:Theory of equations
8934:for satellites at L
7618:casus irreducibilis
7606:Casus irreducibilis
4480:, which are (where
4412:is any of the four
4081:Lagrange resolvents
2191:George Paxton Young
564:Given the equation
9507:Cubic function (3)
9500:Quadratic equation
9230::1:1150–1152 1858.
9084:Harvard University
8911:
8762:
8760:
8390:
8288:
8104:
7914:Francesco Brioschi
7910:elliptic functions
7870:
7689:Francesco Brioschi
7589:
7510:
7472:
7379:
7306:
7265:
7159:
7124:
6946:
6864:
6773:
6693:
6668:
6578:
6503:
6413:
6338:
6251:
6195:
6071:
6008:
5920:
5841:
5762:
5683:
5595:
5534:
5473:
5412:
5335:
5274:
5179:
5154:
5075:
4968:
4914:
4830:
4760:
4690:
4620:
4550:
4470:
4388:
4279:
4226:quadratic equation
4211:
3997:
3860:
3742:
3401:
3237:
3187:
3104:
2969:
2679:
2581:
2405:
2222:Bring–Jerrard form
2172:
2170:
1614:
1612:
1228:Cayley's resolvent
1200:
1198:
779:
659:
517:
349:
147:
28:
9661:
9660:
9602:Quasi-homogeneous
8924:That also yields
8909:
8903:
8667:
8655:
8643:
8625:
8286:
8235:
8071:
8036:
7937:Lagrangian points
7918:Leopold Kronecker
7681:Leopold Kronecker
7653:for real numbers
7509:
7305:
6960:
6959:
6943:
6770:
6692:
6688:
6575:
6410:
6245:
6086:
6085:
5593:
5577:
5532:
5516:
5471:
5455:
5410:
5394:
5266:
5244:
5222:
5206:
5178:
5174:
5073:
5057:
5034:
5011:
4957:
4548:
4383:
4380:
4342:
4112:roots instead of
3985:
3962:
3932:
3854:
3832:
3810:
3788:
3736:
3682:
3631:
3577:
3386:
3356:
3326:
3306:
3182:
3170:
3157:
3155:
3099:
3087:
3077:
3075:
2964:
2895:
2724:The substitution
2672:
2618:
2573:
2509:
2397:
2318:
1194:
1054:
943:
861:
444:Solvable quintics
411:quartic equations
376:quartic equations
184:are members of a
9691:
9524:Quartic equation
9445:
9438:
9431:
9422:
9421:
9379:
9371:
9365:
9357:
9352:. Archived from
9309:
9274:Jörg Bewersdorff
9255::986–992 (1994).
9246:
9219:
9178:
9177:
9165:
9159:
9158:
9146:
9140:
9139:
9127:
9121:
9106:
9100:
9096:David S. Dummit
9094:
9088:
9087:
9079:
9071:
9065:
9059:
9053:
9045::99–130 (1888),
9036:
9030:
9027:
9021:
9013::263-276 (1861)
9004:
8998:
8997:
8987:
8978:
8933:
8920:
8918:
8917:
8912:
8910:
8908:
8902:
8901:
8900:
8887:
8886:
8877:
8876:
8802:
8785:
8771:
8769:
8768:
8763:
8761:
8757:
8756:
8747:
8746:
8727:
8720:
8719:
8707:
8706:
8687:
8677:
8676:
8665:
8653:
8641:
8623:
8622:
8621:
8609:
8608:
8596:
8595:
8573:
8566:
8565:
8550:
8549:
8537:
8536:
8514:
8498:
8497:
8485:
8484:
8462:
8452:
8451:
8439:
8438:
8416:
8399:
8397:
8396:
8391:
8368:
8367:
8352:
8351:
8336:
8335:
8320:
8319:
8297:
8295:
8294:
8289:
8287:
8285:
8284:
8275:
8271:
8270:
8258:
8257:
8241:
8236:
8234:
8233:
8224:
8223:
8222:
8209:
8204:
8203:
8145:angular velocity
8131:, respectively;
8113:
8111:
8110:
8105:
8088:
8087:
8072:
8070:
8069:
8060:
8059:
8058:
8042:
8037:
8035:
8034:
8033:
8011:
8010:
8009:
7993:
7900:
7879:
7877:
7876:
7871:
7862:
7861:
7846:
7845:
7833:
7832:
7823:
7822:
7810:
7809:
7800:
7799:
7787:
7786:
7777:
7776:
7764:
7763:
7743:quartic equation
7718:
7713:septic equations
7710:
7658:
7652:
7598:
7596:
7595:
7590:
7584:
7580:
7575:
7574:
7511:
7502:
7481:
7479:
7478:
7473:
7467:
7466:
7439:
7438:
7426:
7425:
7388:
7386:
7385:
7380:
7374:
7370:
7368:
7367:
7352:
7351:
7339:
7338:
7323:
7322:
7307:
7298:
7274:
7272:
7271:
7266:
7248:
7244:
7233:
7232:
7216:
7215:
7187:
7186:
7168:
7166:
7165:
7160:
7133:
7131:
7130:
7125:
7101:
7100:
7066:
7065:
7050:
7049:
7012:
7011:
6983:
6982:
6956:
6955:
6953:
6952:
6947:
6945:
6944:
6939:
6938:
6937:
6912:
6905:
6900:
6873:
6871:
6870:
6865:
6848:
6847:
6832:
6831:
6816:
6815:
6803:
6802:
6783:
6782:
6780:
6779:
6774:
6772:
6771:
6766:
6765:
6764:
6739:
6732:
6727:
6702:
6700:
6699:
6694:
6690:
6686:
6677:
6675:
6674:
6669:
6652:
6651:
6636:
6635:
6620:
6619:
6607:
6606:
6587:
6585:
6584:
6579:
6577:
6576:
6571:
6570:
6569:
6550:
6543:
6538:
6512:
6510:
6509:
6504:
6487:
6486:
6471:
6470:
6455:
6454:
6442:
6441:
6422:
6420:
6419:
6414:
6412:
6411:
6406:
6405:
6404:
6385:
6378:
6373:
6347:
6345:
6344:
6339:
6325:
6324:
6309:
6308:
6293:
6292:
6280:
6279:
6260:
6258:
6257:
6252:
6250:
6246:
6241:
6230:
6204:
6202:
6201:
6196:
6179:
6178:
6163:
6162:
6147:
6146:
6134:
6133:
6116:
6115:
6110:
6109:
6093:
6080:
6078:
6077:
6072:
6055:
6054:
6039:
6038:
6017:
6015:
6014:
6009:
5989:
5988:
5973:
5972:
5951:
5950:
5929:
5927:
5926:
5921:
5904:
5903:
5888:
5887:
5872:
5871:
5850:
5848:
5847:
5842:
5825:
5824:
5809:
5808:
5793:
5792:
5771:
5769:
5768:
5763:
5746:
5745:
5730:
5729:
5714:
5713:
5692:
5690:
5689:
5684:
5664:
5663:
5648:
5647:
5626:
5625:
5604:
5602:
5601:
5596:
5594:
5586:
5578:
5570:
5565:
5564:
5543:
5541:
5540:
5535:
5533:
5525:
5517:
5509:
5504:
5503:
5482:
5480:
5479:
5474:
5472:
5464:
5456:
5448:
5443:
5442:
5421:
5419:
5418:
5413:
5411:
5403:
5395:
5387:
5382:
5381:
5366:
5365:
5344:
5342:
5341:
5336:
5319:
5318:
5303:
5302:
5283:
5281:
5280:
5275:
5273:
5272:
5267:
5265:
5257:
5251:
5250:
5245:
5243:
5235:
5229:
5228:
5223:
5221:
5213:
5207:
5205:
5197:
5188:
5186:
5185:
5180:
5176:
5172:
5163:
5161:
5160:
5155:
5138:
5137:
5122:
5121:
5106:
5105:
5084:
5082:
5081:
5076:
5074:
5066:
5058:
5050:
5045:
5044:
5035:
5027:
5022:
5021:
5012:
5004:
4999:
4998:
4977:
4975:
4974:
4969:
4958:
4950:
4945:
4944:
4923:
4921:
4920:
4915:
4898:
4897:
4882:
4881:
4866:
4865:
4848:
4847:
4839:
4837:
4836:
4831:
4829:
4828:
4813:
4812:
4803:
4802:
4787:
4786:
4769:
4767:
4766:
4761:
4759:
4758:
4743:
4742:
4733:
4732:
4717:
4716:
4699:
4697:
4696:
4691:
4689:
4688:
4673:
4672:
4663:
4662:
4647:
4646:
4629:
4627:
4626:
4621:
4619:
4618:
4603:
4602:
4593:
4592:
4577:
4576:
4559:
4557:
4556:
4551:
4549:
4544:
4543:
4534:
4529:
4528:
4519:
4518:
4503:
4502:
4479:
4477:
4476:
4471:
4463:
4462:
4447:
4446:
4397:
4395:
4394:
4389:
4384:
4382:
4381:
4379:
4374:
4373:
4364:
4362:
4361:
4348:
4343:
4341:
4336:
4335:
4326:
4324:
4323:
4311:
4310:
4288:
4286:
4285:
4280:
4268:
4267:
4246:
4245:
4220:
4218:
4217:
4212:
4191:
4190:
4175:
4174:
4156:
4155:
4132:
4123:
4117:
4111:
4105:
4100:. However these
4099:
4088:
4078:
4068:
4062:
4056:
4050:
4044:
4037:
4026:
4021:of prime degree
4020:
4006:
4004:
4003:
3998:
3986:
3984:
3983:
3971:
3963:
3961:
3960:
3948:
3943:
3942:
3933:
3925:
3920:
3919:
3904:
3903:
3884:quartic equation
3881:
3869:
3867:
3866:
3861:
3855:
3853:
3848:
3847:
3838:
3833:
3831:
3826:
3825:
3816:
3811:
3809:
3804:
3803:
3794:
3789:
3787:
3782:
3781:
3772:
3751:
3749:
3748:
3743:
3737:
3735:
3730:
3714:
3713:
3688:
3683:
3681:
3676:
3675:
3674:
3637:
3632:
3630:
3625:
3624:
3623:
3583:
3578:
3576:
3571:
3555:
3554:
3532:
3508:
3497:
3496:
3494:
3493:
3490:
3487:
3486:
3485:
3469:
3468:
3467:
3456:
3455:
3454:
3440:
3439:
3438:
3424:
3410:
3408:
3407:
3402:
3397:
3396:
3391:
3387:
3385:
3377:
3367:
3366:
3361:
3357:
3355:
3347:
3337:
3336:
3331:
3327:
3325:
3317:
3307:
3305:
3297:
3273:
3246:
3244:
3243:
3238:
3196:
3194:
3193:
3188:
3183:
3178:
3171:
3166:
3158:
3156:
3151:
3134:
3128:
3113:
3111:
3110:
3105:
3100:
3095:
3088:
3083:
3078:
3076:
3071:
3057:
3054:
3027:
3023:
3019:
3015:
3006:
2978:
2976:
2975:
2970:
2965:
2963:
2962:
2961:
2949:
2948:
2938:
2925:
2924:
2905:
2896:
2894:
2893:
2892:
2880:
2879:
2869:
2856:
2855:
2833:
2815:
2811:
2807:
2792:
2788:
2781:
2774:
2773:
2771:
2770:
2765:
2762:
2750:
2749:
2747:
2746:
2741:
2738:
2720:
2719:
2717:
2716:
2710:
2707:
2688:
2686:
2685:
2680:
2678:
2674:
2673:
2641:
2619:
2611:
2590:
2588:
2587:
2582:
2574:
2572:
2565:
2564:
2554:
2535:
2534:
2518:
2510:
2508:
2501:
2500:
2490:
2471:
2470:
2457:
2452:
2451:
2429:
2423:
2414:
2412:
2411:
2406:
2398:
2396:
2389:
2388:
2378:
2341:
2340:
2327:
2319:
2317:
2310:
2309:
2299:
2280:
2279:
2266:
2261:
2260:
2241:
2219:
2181:
2179:
2178:
2173:
2171:
2164:
2163:
2154:
2153:
2144:
2143:
2125:
2124:
2115:
2114:
2093:
2092:
2074:
2073:
2064:
2063:
2045:
2044:
2029:
2028:
2016:
2015:
1997:
1993:
1992:
1983:
1982:
1973:
1972:
1957:
1956:
1935:
1934:
1925:
1924:
1906:
1905:
1893:
1892:
1877:
1876:
1867:
1866:
1845:
1844:
1823:
1819:
1818:
1809:
1808:
1793:
1792:
1777:
1776:
1767:
1766:
1748:
1747:
1735:
1734:
1710:
1709:
1694:
1693:
1678:
1677:
1668:
1667:
1648:
1623:
1621:
1620:
1615:
1613:
1594:
1593:
1578:
1577:
1556:
1552:
1551:
1536:
1535:
1508:
1507:
1492:
1491:
1482:
1481:
1466:
1465:
1456:
1455:
1437:
1436:
1421:
1420:
1405:
1398:
1397:
1370:
1369:
1354:
1353:
1332:
1331:
1307:
1306:
1282:
1281:
1269:
1268:
1257:
1235:
1230:
1229:
1223:
1209:
1207:
1206:
1201:
1199:
1195:
1193:
1192:
1191:
1178:
1177:
1176:
1158:
1157:
1136:
1135:
1126:
1125:
1104:
1103:
1085:
1084:
1071:
1055:
1053:
1052:
1051:
1038:
1037:
1036:
1018:
1017:
993:
992:
974:
973:
960:
944:
942:
941:
940:
927:
926:
925:
892:
891:
878:
862:
860:
859:
858:
845:
844:
843:
818:
788:
786:
785:
780:
757:
756:
741:
740:
725:
724:
705:
704:
702:
701:
695:
692:
668:
666:
665:
660:
634:
633:
618:
617:
602:
601:
586:
585:
540:solvable quintic
526:
524:
523:
518:
498:
497:
478:
432:
358:
356:
355:
350:
326:
325:
310:
309:
294:
293:
278:
277:
254:quintic equation
251:
244:
229:quartic function
203:
190:rational numbers
188:, typically the
183:
179:
175:
171:
167:
163:
156:
154:
153:
148:
127:
126:
111:
110:
95:
94:
79:
78:
36:quintic function
9699:
9698:
9694:
9693:
9692:
9690:
9689:
9688:
9664:
9663:
9662:
9657:
9606:
9545:
9488:Linear equation
9458:
9449:
9408:Bruce Bartlett:
9386:
9359:
9358:
9350:
9297:
9294:Wayback Machine
9234:
9216:
9187:
9182:
9181:
9176:(I): 1150–1152.
9166:
9162:
9147:
9143:
9128:
9124:
9107:
9103:
9095:
9091:
9077:
9072:
9068:
9060:
9056:
9037:
9033:
9028:
9024:
9005:
9001:
8985:
8979:
8975:
8970:
8957:Septic function
8952:Sextic equation
8948:
8941:
8937:
8925:
8904:
8896:
8892:
8888:
8882:
8878:
8875:
8864:
8861:
8860:
8851:
8847:
8843:
8836:
8826:
8819:
8809:
8794:
8792:
8777:
8759:
8758:
8752:
8748:
8742:
8738:
8725:
8724:
8715:
8711:
8702:
8698:
8685:
8684:
8672:
8668:
8617:
8613:
8604:
8600:
8591:
8587:
8571:
8570:
8561:
8557:
8545:
8541:
8532:
8528:
8512:
8511:
8493:
8489:
8480:
8476:
8460:
8459:
8447:
8443:
8434:
8430:
8413:
8411:
8408:
8407:
8363:
8359:
8347:
8343:
8331:
8327:
8315:
8311:
8306:
8303:
8302:
8280:
8276:
8266:
8262:
8253:
8249:
8242:
8240:
8229:
8225:
8218:
8214:
8210:
8208:
8199:
8195:
8193:
8190:
8189:
8175:
8168:
8157:semi-major axis
8130:
8123:
8083:
8079:
8065:
8061:
8054:
8050:
8043:
8041:
8029:
8025:
8012:
8005:
8001:
7994:
7992:
7990:
7987:
7986:
7981:
7970:
7955:
7948:
7933:
7906:Charles Hermite
7887:
7857:
7853:
7841:
7837:
7828:
7824:
7818:
7814:
7805:
7801:
7795:
7791:
7782:
7778:
7772:
7768:
7759:
7755:
7753:
7750:
7749:
7735:
7729:
7716:
7708:
7673:cubic equations
7665:theta functions
7661:Charles Hermite
7654:
7639:
7628:
7626:Beyond radicals
7613:cubic equations
7611:Analogously to
7609:
7570:
7566:
7516:
7512:
7500:
7492:
7489:
7488:
7462:
7458:
7434:
7430:
7421:
7417:
7399:
7396:
7395:
7363:
7359:
7347:
7343:
7334:
7330:
7318:
7314:
7312:
7308:
7296:
7288:
7285:
7284:
7228:
7224:
7211:
7207:
7202:
7198:
7182:
7178:
7176:
7173:
7172:
7142:
7139:
7138:
7096:
7092:
7061:
7057:
7045:
7041:
7007:
7003:
6978:
6974:
6972:
6969:
6968:
6933:
6929:
6913:
6911:
6907:
6901:
6890:
6884:
6881:
6880:
6878:
6843:
6839:
6827:
6823:
6811:
6807:
6798:
6794:
6792:
6789:
6788:
6760:
6756:
6740:
6738:
6734:
6728:
6717:
6711:
6708:
6707:
6705:
6684:
6681:
6680:
6647:
6643:
6631:
6627:
6615:
6611:
6602:
6598:
6596:
6593:
6592:
6565:
6561:
6551:
6549:
6545:
6539:
6528:
6522:
6519:
6518:
6482:
6478:
6466:
6462:
6450:
6446:
6437:
6433:
6431:
6428:
6427:
6400:
6396:
6386:
6384:
6380:
6374:
6363:
6357:
6354:
6353:
6320:
6316:
6304:
6300:
6288:
6284:
6275:
6271:
6269:
6266:
6265:
6231:
6229:
6225:
6214:
6211:
6210:
6174:
6170:
6158:
6154:
6142:
6138:
6129:
6125:
6123:
6120:
6119:
6100:
6099:
6091:
6050:
6046:
6034:
6030:
6028:
6025:
6024:
5984:
5980:
5968:
5964:
5946:
5942:
5940:
5937:
5936:
5899:
5895:
5883:
5879:
5867:
5863:
5861:
5858:
5857:
5820:
5816:
5804:
5800:
5788:
5784:
5782:
5779:
5778:
5741:
5737:
5725:
5721:
5709:
5705:
5703:
5700:
5699:
5659:
5655:
5643:
5639:
5621:
5617:
5615:
5612:
5611:
5585:
5569:
5560:
5556:
5554:
5551:
5550:
5524:
5508:
5499:
5495:
5493:
5490:
5489:
5463:
5447:
5438:
5434:
5432:
5429:
5428:
5402:
5386:
5377:
5373:
5361:
5357:
5355:
5352:
5351:
5314:
5310:
5298:
5294:
5292:
5289:
5288:
5268:
5261:
5256:
5255:
5246:
5239:
5234:
5233:
5224:
5217:
5212:
5211:
5201:
5196:
5194:
5191:
5190:
5170:
5167:
5166:
5133:
5129:
5117:
5113:
5101:
5097:
5095:
5092:
5091:
5065:
5049:
5040:
5036:
5026:
5017:
5013:
5003:
4994:
4990:
4988:
4985:
4984:
4949:
4940:
4936:
4934:
4931:
4930:
4893:
4889:
4877:
4873:
4861:
4857:
4855:
4852:
4851:
4824:
4820:
4808:
4804:
4798:
4794:
4782:
4778:
4776:
4773:
4772:
4754:
4750:
4738:
4734:
4728:
4724:
4712:
4708:
4706:
4703:
4702:
4684:
4680:
4668:
4664:
4658:
4654:
4642:
4638:
4636:
4633:
4632:
4614:
4610:
4598:
4594:
4588:
4584:
4572:
4568:
4566:
4563:
4562:
4539:
4535:
4533:
4524:
4520:
4514:
4510:
4498:
4494:
4492:
4489:
4488:
4458:
4454:
4442:
4438:
4436:
4433:
4432:
4426:
4406:
4375:
4369:
4365:
4363:
4357:
4353:
4352:
4347:
4337:
4331:
4327:
4325:
4319:
4315:
4306:
4302:
4300:
4297:
4296:
4263:
4259:
4241:
4237:
4235:
4232:
4231:
4186:
4182:
4170:
4166:
4151:
4147:
4145:
4142:
4141:
4128:
4119:
4113:
4107:
4101:
4090:
4084:
4074:
4064:
4058:
4052:
4046:
4039:
4028:
4022:
4011:
3979:
3975:
3970:
3956:
3952:
3947:
3938:
3934:
3924:
3915:
3911:
3899:
3895:
3893:
3890:
3889:
3879:
3874:
3849:
3843:
3839:
3837:
3827:
3821:
3817:
3815:
3805:
3799:
3795:
3793:
3783:
3777:
3773:
3771:
3763:
3760:
3759:
3731:
3709:
3705:
3689:
3687:
3677:
3670:
3666:
3638:
3636:
3626:
3619:
3615:
3584:
3582:
3572:
3550:
3546:
3533:
3531:
3517:
3514:
3513:
3503:
3491:
3488:
3483:
3481:
3479:
3478:
3476:
3471:
3465:
3463:
3458:
3449:
3447:
3442:
3433:
3431:
3426:
3415:
3392:
3381:
3376:
3372:
3371:
3362:
3351:
3346:
3342:
3341:
3332:
3321:
3316:
3312:
3311:
3301:
3296:
3282:
3279:
3278:
3252:
3205:
3202:
3201:
3165:
3150:
3133:
3129:
3127:
3125:
3122:
3121:
3082:
3070:
3056:
3055:
3053:
3051:
3048:
3047:
3025:
3021:
3017:
3014:
3008:
3005:
2999:
2997:symmetric group
2985:
2957:
2953:
2944:
2940:
2939:
2920:
2916:
2906:
2904:
2888:
2884:
2875:
2871:
2870:
2851:
2847:
2834:
2832:
2824:
2821:
2820:
2813:
2809:
2794:
2790:
2786:
2776:
2766:
2763:
2760:
2759:
2757:
2752:
2742:
2739:
2733:
2732:
2730:
2725:
2711:
2708:
2701:
2700:
2698:
2693:
2640:
2624:
2620:
2610:
2602:
2599:
2598:
2560:
2556:
2555:
2530:
2526:
2519:
2517:
2496:
2492:
2491:
2466:
2462:
2458:
2456:
2447:
2443:
2441:
2438:
2437:
2425:
2419:
2384:
2380:
2379:
2336:
2332:
2328:
2326:
2305:
2301:
2300:
2275:
2271:
2267:
2265:
2256:
2252:
2250:
2247:
2246:
2236:
2206:
2203:
2169:
2168:
2159:
2155:
2149:
2145:
2139:
2135:
2120:
2116:
2110:
2106:
2088:
2084:
2069:
2065:
2059:
2055:
2040:
2036:
2024:
2020:
2011:
2007:
1995:
1994:
1988:
1984:
1978:
1974:
1968:
1964:
1952:
1948:
1930:
1926:
1920:
1916:
1901:
1897:
1888:
1884:
1872:
1868:
1862:
1858:
1840:
1836:
1821:
1820:
1814:
1810:
1804:
1800:
1788:
1784:
1772:
1768:
1762:
1758:
1743:
1739:
1730:
1726:
1705:
1701:
1689:
1685:
1673:
1669:
1663:
1659:
1649:
1647:
1637:
1635:
1632:
1631:
1611:
1610:
1589:
1585:
1573:
1569:
1554:
1553:
1547:
1543:
1531:
1527:
1503:
1499:
1487:
1483:
1477:
1473:
1461:
1457:
1451:
1447:
1432:
1428:
1416:
1412:
1403:
1402:
1393:
1389:
1365:
1361:
1349:
1345:
1327:
1323:
1302:
1298:
1277:
1273:
1264:
1260:
1258:
1256:
1246:
1244:
1241:
1240:
1233:
1227:
1226:
1214:
1197:
1196:
1187:
1183:
1179:
1172:
1168:
1153:
1149:
1131:
1127:
1121:
1117:
1099:
1095:
1080:
1076:
1072:
1070:
1063:
1057:
1056:
1047:
1043:
1039:
1032:
1028:
1013:
1009:
988:
984:
969:
965:
961:
959:
952:
946:
945:
936:
932:
928:
921:
917:
887:
883:
879:
877:
870:
864:
863:
854:
850:
846:
839:
835:
819:
817:
810:
803:
801:
798:
797:
752:
748:
736:
732:
720:
716:
714:
711:
710:
696:
693:
688:
687:
685:
676:
629:
625:
613:
609:
597:
593:
581:
577:
572:
569:
568:
547:Évariste Galois
493:
489:
487:
484:
483:
453:
446:
423:
388:
321:
317:
305:
301:
289:
285:
273:
269:
264:
261:
260:
246:
235:
217:cubic functions
201:
198:complex numbers
181:
177:
173:
169:
165:
161:
122:
118:
106:
102:
90:
86:
74:
70:
50:
47:
46:
25:critical points
17:
12:
11:
5:
9697:
9687:
9686:
9681:
9676:
9659:
9658:
9656:
9655:
9650:
9645:
9640:
9635:
9630:
9625:
9620:
9614:
9612:
9608:
9607:
9605:
9604:
9599:
9594:
9589:
9584:
9579:
9574:
9569:
9564:
9559:
9553:
9551:
9547:
9546:
9544:
9543:
9538:
9533:
9528:
9527:
9526:
9516:
9515:
9514:
9512:Cubic equation
9504:
9503:
9502:
9492:
9491:
9490:
9480:
9475:
9469:
9467:
9460:
9459:
9448:
9447:
9440:
9433:
9425:
9419:
9418:
9405:
9399:
9393:
9385:
9384:External links
9382:
9381:
9380:
9372:
9348:
9325:
9318:
9311:
9271:
9256:
9231:
9220:
9214:
9199:
9186:
9183:
9180:
9179:
9160:
9141:
9122:
9101:
9089:
9074:Elkies, Noam.
9066:
9064:, p. 207)
9054:
9031:
9022:
8999:
8972:
8971:
8969:
8966:
8965:
8964:
8959:
8954:
8947:
8944:
8939:
8935:
8922:
8921:
8907:
8899:
8895:
8891:
8885:
8881:
8874:
8871:
8868:
8849:
8845:
8841:
8834:
8824:
8817:
8807:
8790:
8774:
8773:
8755:
8751:
8745:
8741:
8737:
8734:
8731:
8728:
8726:
8723:
8718:
8714:
8710:
8705:
8701:
8697:
8694:
8691:
8688:
8686:
8683:
8680:
8675:
8671:
8664:
8661:
8658:
8652:
8649:
8646:
8640:
8637:
8634:
8631:
8628:
8620:
8616:
8612:
8607:
8603:
8599:
8594:
8590:
8586:
8583:
8580:
8577:
8574:
8572:
8569:
8564:
8560:
8556:
8553:
8548:
8544:
8540:
8535:
8531:
8527:
8524:
8521:
8518:
8515:
8513:
8510:
8507:
8504:
8501:
8496:
8492:
8488:
8483:
8479:
8475:
8472:
8469:
8466:
8463:
8461:
8458:
8455:
8450:
8446:
8442:
8437:
8433:
8429:
8426:
8423:
8420:
8417:
8415:
8401:
8400:
8389:
8386:
8383:
8380:
8377:
8374:
8371:
8366:
8362:
8358:
8355:
8350:
8346:
8342:
8339:
8334:
8330:
8326:
8323:
8318:
8314:
8310:
8283:
8279:
8274:
8269:
8265:
8261:
8256:
8252:
8248:
8245:
8239:
8232:
8228:
8221:
8217:
8213:
8207:
8202:
8198:
8173:
8166:
8128:
8121:
8115:
8114:
8103:
8100:
8097:
8094:
8091:
8086:
8082:
8078:
8075:
8068:
8064:
8057:
8053:
8049:
8046:
8040:
8032:
8028:
8024:
8021:
8018:
8015:
8008:
8004:
8000:
7997:
7979:
7968:
7953:
7946:
7932:
7929:
7881:
7880:
7868:
7865:
7860:
7856:
7852:
7849:
7844:
7840:
7836:
7831:
7827:
7821:
7817:
7813:
7808:
7804:
7798:
7794:
7790:
7785:
7781:
7775:
7771:
7767:
7762:
7758:
7731:Main article:
7728:
7725:
7627:
7624:
7608:
7603:
7602:
7601:
7600:
7599:
7588:
7583:
7578:
7573:
7569:
7565:
7562:
7559:
7556:
7553:
7550:
7547:
7544:
7541:
7538:
7535:
7532:
7529:
7526:
7523:
7520:
7515:
7508:
7505:
7499:
7496:
7484:
7483:
7482:
7471:
7465:
7461:
7457:
7454:
7451:
7448:
7445:
7442:
7437:
7433:
7429:
7424:
7420:
7416:
7413:
7409:
7406:
7403:
7391:
7390:
7389:
7378:
7373:
7366:
7362:
7358:
7355:
7350:
7346:
7342:
7337:
7333:
7329:
7326:
7321:
7317:
7311:
7304:
7301:
7295:
7292:
7276:
7275:
7264:
7261:
7258:
7254:
7251:
7247:
7243:
7239:
7236:
7231:
7227:
7222:
7219:
7214:
7210:
7205:
7201:
7197:
7193:
7190:
7185:
7181:
7158:
7155:
7152:
7149:
7146:
7135:
7134:
7123:
7120:
7117:
7114:
7111:
7107:
7104:
7099:
7095:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7064:
7060:
7056:
7053:
7048:
7044:
7039:
7036:
7033:
7030:
7027:
7024:
7021:
7018:
7015:
7010:
7006:
7001:
6998:
6995:
6992:
6989:
6986:
6981:
6977:
6962:
6961:
6958:
6957:
6942:
6936:
6932:
6928:
6925:
6922:
6919:
6916:
6910:
6904:
6899:
6896:
6893:
6889:
6876:
6874:
6863:
6860:
6857:
6854:
6851:
6846:
6842:
6838:
6835:
6830:
6826:
6822:
6819:
6814:
6810:
6806:
6801:
6797:
6785:
6784:
6769:
6763:
6759:
6755:
6752:
6749:
6746:
6743:
6737:
6731:
6726:
6723:
6720:
6716:
6703:
6678:
6667:
6664:
6661:
6658:
6655:
6650:
6646:
6642:
6639:
6634:
6630:
6626:
6623:
6618:
6614:
6610:
6605:
6601:
6589:
6588:
6574:
6568:
6564:
6560:
6557:
6554:
6548:
6542:
6537:
6534:
6531:
6527:
6515:
6513:
6502:
6499:
6496:
6493:
6490:
6485:
6481:
6477:
6474:
6469:
6465:
6461:
6458:
6453:
6449:
6445:
6440:
6436:
6424:
6423:
6409:
6403:
6399:
6395:
6392:
6389:
6383:
6377:
6372:
6369:
6366:
6362:
6350:
6348:
6337:
6334:
6331:
6328:
6323:
6319:
6315:
6312:
6307:
6303:
6299:
6296:
6291:
6287:
6283:
6278:
6274:
6262:
6261:
6249:
6244:
6240:
6237:
6234:
6228:
6224:
6221:
6218:
6207:
6205:
6194:
6191:
6188:
6185:
6182:
6177:
6173:
6169:
6166:
6161:
6157:
6153:
6150:
6145:
6141:
6137:
6132:
6128:
6096:roots of unity
6088:
6087:
6084:
6083:
6081:
6070:
6067:
6064:
6061:
6058:
6053:
6049:
6045:
6042:
6037:
6033:
6021:
6020:
6018:
6007:
6004:
6001:
5998:
5995:
5992:
5987:
5983:
5979:
5976:
5971:
5967:
5963:
5960:
5957:
5954:
5949:
5945:
5933:
5932:
5930:
5919:
5916:
5913:
5910:
5907:
5902:
5898:
5894:
5891:
5886:
5882:
5878:
5875:
5870:
5866:
5854:
5853:
5851:
5840:
5837:
5834:
5831:
5828:
5823:
5819:
5815:
5812:
5807:
5803:
5799:
5796:
5791:
5787:
5775:
5774:
5772:
5761:
5758:
5755:
5752:
5749:
5744:
5740:
5736:
5733:
5728:
5724:
5720:
5717:
5712:
5708:
5696:
5695:
5693:
5682:
5679:
5676:
5673:
5670:
5667:
5662:
5658:
5654:
5651:
5646:
5642:
5638:
5635:
5632:
5629:
5624:
5620:
5608:
5607:
5605:
5592:
5589:
5584:
5581:
5576:
5573:
5568:
5563:
5559:
5547:
5546:
5544:
5531:
5528:
5523:
5520:
5515:
5512:
5507:
5502:
5498:
5486:
5485:
5483:
5470:
5467:
5462:
5459:
5454:
5451:
5446:
5441:
5437:
5425:
5424:
5422:
5409:
5406:
5401:
5398:
5393:
5390:
5385:
5380:
5376:
5372:
5369:
5364:
5360:
5348:
5347:
5345:
5334:
5331:
5328:
5325:
5322:
5317:
5313:
5309:
5306:
5301:
5297:
5285:
5284:
5271:
5264:
5260:
5254:
5249:
5242:
5238:
5232:
5227:
5220:
5216:
5210:
5204:
5200:
5164:
5153:
5150:
5147:
5144:
5141:
5136:
5132:
5128:
5125:
5120:
5116:
5112:
5109:
5104:
5100:
5088:
5087:
5085:
5072:
5069:
5064:
5061:
5056:
5053:
5048:
5043:
5039:
5033:
5030:
5025:
5020:
5016:
5010:
5007:
5002:
4997:
4993:
4981:
4980:
4978:
4967:
4964:
4961:
4956:
4953:
4948:
4943:
4939:
4927:
4926:
4924:
4913:
4910:
4907:
4904:
4901:
4896:
4892:
4888:
4885:
4880:
4876:
4872:
4869:
4864:
4860:
4841:
4840:
4827:
4823:
4819:
4816:
4811:
4807:
4801:
4797:
4793:
4790:
4785:
4781:
4770:
4757:
4753:
4749:
4746:
4741:
4737:
4731:
4727:
4723:
4720:
4715:
4711:
4700:
4687:
4683:
4679:
4676:
4671:
4667:
4661:
4657:
4653:
4650:
4645:
4641:
4630:
4617:
4613:
4609:
4606:
4601:
4597:
4591:
4587:
4583:
4580:
4575:
4571:
4560:
4547:
4542:
4538:
4532:
4527:
4523:
4517:
4513:
4509:
4506:
4501:
4497:
4469:
4466:
4461:
4457:
4453:
4450:
4445:
4441:
4425:
4422:
4404:
4399:
4398:
4387:
4378:
4372:
4368:
4360:
4356:
4351:
4346:
4340:
4334:
4330:
4322:
4318:
4314:
4309:
4305:
4290:
4289:
4278:
4274:
4271:
4266:
4262:
4258:
4255:
4252:
4249:
4244:
4240:
4222:
4221:
4210:
4206:
4203:
4200:
4197:
4194:
4189:
4185:
4181:
4178:
4173:
4169:
4165:
4162:
4159:
4154:
4150:
4051:is the sum of
4008:
4007:
3996:
3992:
3989:
3982:
3978:
3974:
3969:
3966:
3959:
3955:
3951:
3946:
3941:
3937:
3931:
3928:
3923:
3918:
3914:
3910:
3907:
3902:
3898:
3877:
3871:
3870:
3859:
3852:
3846:
3842:
3836:
3830:
3824:
3820:
3814:
3808:
3802:
3798:
3792:
3786:
3780:
3776:
3770:
3767:
3753:
3752:
3741:
3734:
3729:
3726:
3723:
3720:
3717:
3712:
3708:
3704:
3701:
3698:
3695:
3692:
3686:
3680:
3673:
3669:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3635:
3629:
3622:
3618:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3581:
3575:
3570:
3567:
3564:
3561:
3558:
3553:
3549:
3545:
3542:
3539:
3536:
3530:
3527:
3524:
3521:
3412:
3411:
3400:
3395:
3390:
3384:
3380:
3375:
3370:
3365:
3360:
3354:
3350:
3345:
3340:
3335:
3330:
3324:
3320:
3315:
3310:
3304:
3300:
3295:
3292:
3289:
3286:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3198:
3197:
3186:
3181:
3177:
3174:
3169:
3164:
3161:
3154:
3149:
3146:
3143:
3140:
3137:
3132:
3115:
3114:
3103:
3098:
3094:
3091:
3086:
3081:
3074:
3069:
3066:
3063:
3060:
3012:
3003:
2993:solvable group
2984:
2981:
2980:
2979:
2968:
2960:
2956:
2952:
2947:
2943:
2937:
2934:
2931:
2928:
2923:
2919:
2915:
2912:
2909:
2903:
2900:
2891:
2887:
2883:
2878:
2874:
2868:
2865:
2862:
2859:
2854:
2850:
2846:
2843:
2840:
2837:
2831:
2828:
2690:
2689:
2677:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2639:
2636:
2633:
2630:
2627:
2623:
2617:
2614:
2609:
2606:
2592:
2591:
2580:
2577:
2571:
2568:
2563:
2559:
2553:
2550:
2547:
2544:
2541:
2538:
2533:
2529:
2525:
2522:
2516:
2513:
2507:
2504:
2499:
2495:
2489:
2486:
2483:
2480:
2477:
2474:
2469:
2465:
2461:
2455:
2450:
2446:
2430:are rational.
2416:
2415:
2404:
2401:
2395:
2392:
2387:
2383:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2339:
2335:
2331:
2325:
2322:
2316:
2313:
2308:
2304:
2298:
2295:
2292:
2289:
2286:
2283:
2278:
2274:
2270:
2264:
2259:
2255:
2202:
2199:
2183:
2182:
2167:
2162:
2158:
2152:
2148:
2142:
2138:
2134:
2131:
2128:
2123:
2119:
2113:
2109:
2105:
2102:
2099:
2096:
2091:
2087:
2083:
2080:
2077:
2072:
2068:
2062:
2058:
2054:
2051:
2048:
2043:
2039:
2035:
2032:
2027:
2023:
2019:
2014:
2010:
2006:
2003:
2000:
1998:
1996:
1991:
1987:
1981:
1977:
1971:
1967:
1963:
1960:
1955:
1951:
1947:
1944:
1941:
1938:
1933:
1929:
1923:
1919:
1915:
1912:
1909:
1904:
1900:
1896:
1891:
1887:
1883:
1880:
1875:
1871:
1865:
1861:
1857:
1854:
1851:
1848:
1843:
1839:
1835:
1832:
1829:
1826:
1824:
1822:
1817:
1813:
1807:
1803:
1799:
1796:
1791:
1787:
1783:
1780:
1775:
1771:
1765:
1761:
1757:
1754:
1751:
1746:
1742:
1738:
1733:
1729:
1725:
1722:
1719:
1716:
1713:
1708:
1704:
1700:
1697:
1692:
1688:
1684:
1681:
1676:
1672:
1666:
1662:
1658:
1655:
1652:
1650:
1646:
1643:
1640:
1639:
1625:
1624:
1609:
1606:
1603:
1600:
1597:
1592:
1588:
1584:
1581:
1576:
1572:
1568:
1565:
1562:
1559:
1557:
1555:
1550:
1546:
1542:
1539:
1534:
1530:
1526:
1523:
1520:
1517:
1514:
1511:
1506:
1502:
1498:
1495:
1490:
1486:
1480:
1476:
1472:
1469:
1464:
1460:
1454:
1450:
1446:
1443:
1440:
1435:
1431:
1427:
1424:
1419:
1415:
1411:
1408:
1406:
1404:
1401:
1396:
1392:
1388:
1385:
1382:
1379:
1376:
1373:
1368:
1364:
1360:
1357:
1352:
1348:
1344:
1341:
1338:
1335:
1330:
1326:
1322:
1319:
1316:
1313:
1310:
1305:
1301:
1297:
1294:
1291:
1288:
1285:
1280:
1276:
1272:
1267:
1263:
1259:
1255:
1252:
1249:
1248:
1211:
1210:
1190:
1186:
1182:
1175:
1171:
1167:
1164:
1161:
1156:
1152:
1148:
1145:
1142:
1139:
1134:
1130:
1124:
1120:
1116:
1113:
1110:
1107:
1102:
1098:
1094:
1091:
1088:
1083:
1079:
1075:
1069:
1066:
1064:
1062:
1059:
1058:
1050:
1046:
1042:
1035:
1031:
1027:
1024:
1021:
1016:
1012:
1008:
1005:
1002:
999:
996:
991:
987:
983:
980:
977:
972:
968:
964:
958:
955:
953:
951:
948:
947:
939:
935:
931:
924:
920:
916:
913:
910:
907:
904:
901:
898:
895:
890:
886:
882:
876:
873:
871:
869:
866:
865:
857:
853:
849:
842:
838:
834:
831:
828:
825:
822:
816:
813:
811:
809:
806:
805:
791:
790:
778:
775:
772:
769:
766:
763:
760:
755:
751:
747:
744:
739:
735:
731:
728:
723:
719:
670:
669:
658:
655:
652:
649:
646:
643:
640:
637:
632:
628:
624:
621:
616:
612:
608:
605:
600:
596:
592:
589:
584:
580:
576:
528:
527:
516:
513:
510:
507:
504:
501:
496:
492:
445:
442:
387:
384:
360:
359:
347:
344:
341:
338:
335:
332:
329:
324:
320:
316:
313:
308:
304:
300:
297:
292:
288:
284:
281:
276:
272:
268:
158:
157:
145:
142:
139:
136:
133:
130:
125:
121:
117:
114:
109:
105:
101:
98:
93:
89:
85:
82:
77:
73:
69:
66:
63:
60:
57:
54:
15:
9:
6:
4:
3:
2:
9696:
9685:
9682:
9680:
9679:Galois theory
9677:
9675:
9672:
9671:
9669:
9654:
9653:Gröbner basis
9651:
9649:
9646:
9644:
9641:
9639:
9636:
9634:
9631:
9629:
9626:
9624:
9621:
9619:
9618:Factorization
9616:
9615:
9613:
9609:
9603:
9600:
9598:
9595:
9593:
9590:
9588:
9585:
9583:
9580:
9578:
9575:
9573:
9570:
9568:
9565:
9563:
9560:
9558:
9555:
9554:
9552:
9550:By properties
9548:
9542:
9539:
9537:
9534:
9532:
9529:
9525:
9522:
9521:
9520:
9517:
9513:
9510:
9509:
9508:
9505:
9501:
9498:
9497:
9496:
9493:
9489:
9486:
9485:
9484:
9481:
9479:
9476:
9474:
9471:
9470:
9468:
9466:
9461:
9457:
9453:
9446:
9441:
9439:
9434:
9432:
9427:
9426:
9423:
9416:
9412:
9411:
9406:
9403:
9400:
9397:
9394:
9391:
9388:
9387:
9378:
9373:
9369:
9363:
9355:
9351:
9349:3-540-43826-2
9345:
9341:
9340:
9335:
9331:
9326:
9323:
9319:
9316:
9312:
9308:
9304:
9300:
9295:
9291:
9288:. Chapter 8 (
9287:
9286:0-8218-3817-2
9283:
9279:
9275:
9272:
9269:
9268:0-412-34550-1
9265:
9261:
9260:Galois Theory
9258:Ian Stewart,
9257:
9254:
9250:
9245:
9241:
9237:
9232:
9229:
9225:
9221:
9217:
9215:0-486-49528-0
9211:
9207:
9206:
9200:
9197:
9193:
9189:
9188:
9175:
9171:
9164:
9156:
9152:
9145:
9138:(I): 508–515.
9137:
9133:
9126:
9119:
9115:
9111:
9105:
9099:
9093:
9085:
9081:
9078:a x + b x + c
9070:
9063:
9058:
9052:
9048:
9044:
9041:
9035:
9026:
9020:
9016:
9012:
9009:
9003:
8996:(3): 282–286.
8995:
8991:
8984:
8977:
8973:
8963:
8960:
8958:
8955:
8953:
8950:
8949:
8943:
8932:
8928:
8905:
8897:
8893:
8889:
8883:
8879:
8872:
8869:
8866:
8859:
8858:
8857:
8855:
8840:
8833:
8828:
8823:
8816:
8813:
8806:
8801:
8798:= 1.491 x 10
8797:
8789:
8784:
8781:= 1.501 x 10
8780:
8753:
8749:
8743:
8739:
8735:
8732:
8729:
8721:
8716:
8712:
8708:
8703:
8699:
8695:
8692:
8689:
8681:
8673:
8669:
8662:
8659:
8656:
8650:
8647:
8644:
8638:
8635:
8632:
8629:
8618:
8614:
8605:
8601:
8597:
8592:
8588:
8581:
8578:
8575:
8567:
8562:
8558:
8554:
8546:
8542:
8538:
8533:
8529:
8522:
8519:
8516:
8508:
8505:
8502:
8494:
8490:
8486:
8481:
8477:
8470:
8467:
8464:
8456:
8448:
8444:
8440:
8435:
8431:
8424:
8421:
8418:
8406:
8405:
8404:
8387:
8384:
8381:
8378:
8375:
8372:
8369:
8364:
8360:
8356:
8353:
8348:
8344:
8340:
8337:
8332:
8328:
8324:
8321:
8316:
8312:
8308:
8301:
8300:
8299:
8281:
8277:
8267:
8263:
8259:
8254:
8250:
8243:
8237:
8230:
8226:
8219:
8215:
8211:
8205:
8200:
8196:
8186:
8184:
8180:
8176:
8169:
8162:
8158:
8154:
8150:
8146:
8142:
8138:
8134:
8127:
8120:
8098:
8095:
8092:
8084:
8080:
8076:
8073:
8066:
8062:
8055:
8051:
8047:
8044:
8038:
8030:
8022:
8019:
8016:
8006:
8002:
7998:
7995:
7985:
7984:
7983:
7978:
7974:
7967:
7963:
7959:
7952:
7945:
7940:
7938:
7928:
7926:
7925:Bring radical
7921:
7919:
7915:
7911:
7907:
7902:
7898:
7894:
7890:
7886:
7866:
7863:
7858:
7854:
7850:
7847:
7842:
7838:
7834:
7829:
7825:
7819:
7815:
7811:
7806:
7802:
7796:
7792:
7788:
7783:
7779:
7773:
7769:
7765:
7760:
7756:
7748:
7747:
7746:
7744:
7740:
7734:
7733:Bring radical
7724:
7722:
7714:
7706:
7702:
7701:Galois theory
7698:
7694:
7690:
7686:
7682:
7678:
7674:
7670:
7666:
7662:
7657:
7650:
7646:
7642:
7637:
7636:ultraradicals
7633:
7623:
7620:
7619:
7614:
7607:
7586:
7581:
7576:
7571:
7567:
7563:
7557:
7554:
7551:
7548:
7542:
7539:
7533:
7530:
7527:
7524:
7518:
7513:
7506:
7503:
7497:
7494:
7487:
7486:
7485:
7469:
7463:
7459:
7455:
7452:
7449:
7446:
7443:
7435:
7431:
7427:
7422:
7418:
7414:
7407:
7404:
7401:
7394:
7393:
7392:
7376:
7371:
7364:
7360:
7356:
7348:
7344:
7340:
7335:
7331:
7327:
7319:
7315:
7309:
7302:
7299:
7293:
7290:
7283:
7282:
7281:
7280:
7279:
7262:
7259:
7256:
7252:
7249:
7245:
7241:
7237:
7234:
7229:
7225:
7220:
7217:
7212:
7208:
7203:
7199:
7195:
7191:
7188:
7183:
7179:
7171:
7170:
7169:
7156:
7153:
7150:
7147:
7144:
7121:
7118:
7115:
7112:
7109:
7105:
7102:
7097:
7093:
7085:
7082:
7079:
7076:
7073:
7070:
7067:
7062:
7058:
7051:
7046:
7042:
7034:
7031:
7028:
7025:
7022:
7019:
7013:
7008:
7004:
6996:
6993:
6990:
6984:
6979:
6975:
6967:
6966:
6965:
6940:
6934:
6926:
6920:
6917:
6914:
6908:
6902:
6897:
6894:
6891:
6887:
6877:
6875:
6861:
6858:
6855:
6852:
6849:
6844:
6840:
6836:
6833:
6828:
6824:
6820:
6817:
6812:
6808:
6804:
6799:
6795:
6787:
6786:
6767:
6761:
6753:
6747:
6744:
6741:
6735:
6729:
6724:
6721:
6718:
6714:
6704:
6679:
6665:
6662:
6659:
6656:
6653:
6648:
6644:
6640:
6637:
6632:
6628:
6624:
6621:
6616:
6612:
6608:
6603:
6599:
6591:
6590:
6572:
6566:
6562:
6558:
6555:
6552:
6546:
6540:
6535:
6532:
6529:
6525:
6516:
6514:
6500:
6497:
6494:
6491:
6488:
6483:
6479:
6475:
6472:
6467:
6463:
6459:
6456:
6451:
6447:
6443:
6438:
6434:
6426:
6425:
6407:
6401:
6397:
6393:
6390:
6387:
6381:
6375:
6370:
6367:
6364:
6360:
6351:
6349:
6335:
6332:
6329:
6326:
6321:
6317:
6313:
6310:
6305:
6301:
6297:
6294:
6289:
6285:
6281:
6276:
6272:
6264:
6263:
6247:
6242:
6238:
6235:
6232:
6226:
6222:
6219:
6216:
6208:
6206:
6192:
6189:
6186:
6183:
6180:
6175:
6171:
6167:
6164:
6159:
6155:
6151:
6148:
6143:
6139:
6135:
6130:
6126:
6118:
6117:
6114:
6113:
6112:
6107:
6103:
6097:
6082:
6068:
6065:
6062:
6059:
6056:
6051:
6047:
6043:
6040:
6035:
6031:
6023:
6022:
6019:
6002:
5999:
5996:
5993:
5990:
5985:
5981:
5977:
5974:
5969:
5965:
5961:
5955:
5952:
5947:
5943:
5935:
5934:
5931:
5917:
5914:
5911:
5908:
5905:
5900:
5896:
5892:
5889:
5884:
5880:
5876:
5873:
5868:
5864:
5856:
5855:
5852:
5838:
5835:
5832:
5829:
5826:
5821:
5817:
5813:
5810:
5805:
5801:
5797:
5794:
5789:
5785:
5777:
5776:
5773:
5759:
5756:
5753:
5750:
5747:
5742:
5738:
5734:
5731:
5726:
5722:
5718:
5715:
5710:
5706:
5698:
5697:
5694:
5677:
5674:
5671:
5668:
5665:
5660:
5656:
5652:
5649:
5644:
5640:
5636:
5630:
5627:
5622:
5618:
5610:
5609:
5606:
5590:
5587:
5582:
5579:
5574:
5571:
5566:
5561:
5557:
5549:
5548:
5545:
5529:
5526:
5521:
5518:
5513:
5510:
5505:
5500:
5496:
5488:
5487:
5484:
5468:
5465:
5460:
5457:
5452:
5449:
5444:
5439:
5435:
5427:
5426:
5423:
5407:
5404:
5399:
5396:
5391:
5388:
5383:
5378:
5374:
5370:
5367:
5362:
5358:
5350:
5349:
5346:
5332:
5329:
5326:
5323:
5320:
5315:
5311:
5307:
5304:
5299:
5295:
5287:
5286:
5269:
5262:
5258:
5252:
5247:
5240:
5236:
5230:
5225:
5218:
5214:
5208:
5202:
5198:
5165:
5151:
5148:
5145:
5142:
5139:
5134:
5130:
5126:
5123:
5118:
5114:
5110:
5107:
5102:
5098:
5090:
5089:
5086:
5070:
5067:
5062:
5059:
5054:
5051:
5046:
5041:
5037:
5031:
5028:
5023:
5018:
5014:
5008:
5005:
5000:
4995:
4991:
4983:
4982:
4979:
4965:
4962:
4959:
4954:
4951:
4946:
4941:
4937:
4929:
4928:
4925:
4911:
4908:
4905:
4902:
4899:
4894:
4890:
4886:
4883:
4878:
4874:
4870:
4867:
4862:
4858:
4850:
4849:
4846:
4845:
4844:
4825:
4821:
4817:
4814:
4809:
4805:
4799:
4795:
4791:
4788:
4783:
4779:
4771:
4755:
4751:
4747:
4744:
4739:
4735:
4729:
4725:
4721:
4718:
4713:
4709:
4701:
4685:
4681:
4677:
4674:
4669:
4665:
4659:
4655:
4651:
4648:
4643:
4639:
4631:
4615:
4611:
4607:
4604:
4599:
4595:
4589:
4585:
4581:
4578:
4573:
4569:
4561:
4545:
4540:
4536:
4530:
4525:
4521:
4515:
4511:
4507:
4504:
4499:
4495:
4487:
4486:
4485:
4483:
4467:
4464:
4459:
4455:
4451:
4448:
4443:
4439:
4429:
4421:
4419:
4415:
4411:
4407:
4385:
4376:
4370:
4366:
4358:
4354:
4349:
4344:
4338:
4332:
4328:
4320:
4316:
4312:
4307:
4303:
4295:
4294:
4293:
4276:
4272:
4269:
4264:
4260:
4256:
4253:
4250:
4247:
4242:
4238:
4230:
4229:
4228:
4227:
4208:
4204:
4201:
4198:
4195:
4192:
4187:
4183:
4179:
4176:
4171:
4167:
4163:
4160:
4157:
4152:
4148:
4140:
4139:
4138:
4136:
4131:
4125:
4122:
4116:
4110:
4104:
4097:
4093:
4087:
4082:
4077:
4072:
4067:
4061:
4055:
4049:
4042:
4035:
4031:
4025:
4018:
4014:
3994:
3990:
3987:
3980:
3976:
3972:
3967:
3964:
3957:
3953:
3949:
3944:
3939:
3935:
3929:
3926:
3921:
3916:
3912:
3908:
3905:
3900:
3896:
3888:
3887:
3886:
3885:
3880:
3857:
3850:
3844:
3840:
3834:
3828:
3822:
3818:
3812:
3806:
3800:
3796:
3790:
3784:
3778:
3774:
3768:
3765:
3758:
3757:
3756:
3739:
3732:
3724:
3721:
3718:
3710:
3702:
3699:
3696:
3693:
3684:
3678:
3671:
3663:
3660:
3657:
3648:
3645:
3642:
3633:
3627:
3620:
3612:
3609:
3606:
3597:
3594:
3591:
3588:
3579:
3573:
3565:
3562:
3559:
3551:
3543:
3540:
3537:
3528:
3525:
3522:
3519:
3512:
3511:
3510:
3507:= −1.84208...
3506:
3501:
3474:
3461:
3453:
3445:
3437:
3429:
3422:
3418:
3398:
3393:
3388:
3382:
3378:
3373:
3368:
3363:
3358:
3352:
3348:
3343:
3338:
3333:
3328:
3322:
3318:
3313:
3308:
3302:
3298:
3293:
3290:
3287:
3284:
3277:
3276:
3275:
3271:
3267:
3263:
3259:
3255:
3248:
3231:
3228:
3225:
3222:
3216:
3213:
3210:
3207:
3184:
3179:
3175:
3172:
3167:
3162:
3159:
3152:
3147:
3144:
3141:
3138:
3135:
3130:
3120:
3119:
3118:
3101:
3096:
3092:
3089:
3084:
3079:
3072:
3067:
3064:
3061:
3058:
3046:
3045:
3044:
3042:
3038:
3034:
3029:
3011:
3002:
2998:
2994:
2990:
2966:
2958:
2954:
2950:
2945:
2941:
2932:
2929:
2926:
2921:
2917:
2913:
2907:
2901:
2898:
2889:
2885:
2881:
2876:
2872:
2863:
2860:
2857:
2852:
2848:
2844:
2838:
2835:
2829:
2826:
2819:
2818:
2817:
2805:
2801:
2797:
2783:
2779:
2769:
2755:
2745:
2737:
2728:
2722:
2714:
2705:
2696:
2675:
2666:
2663:
2660:
2651:
2648:
2645:
2637:
2634:
2631:
2628:
2625:
2621:
2615:
2612:
2607:
2604:
2597:
2596:
2595:
2578:
2575:
2569:
2566:
2561:
2557:
2548:
2545:
2542:
2539:
2531:
2527:
2523:
2520:
2514:
2511:
2505:
2502:
2497:
2493:
2484:
2481:
2478:
2475:
2467:
2463:
2459:
2453:
2448:
2444:
2436:
2435:
2434:
2431:
2428:
2422:
2402:
2399:
2393:
2390:
2385:
2381:
2372:
2369:
2366:
2363:
2354:
2351:
2348:
2345:
2337:
2333:
2329:
2323:
2320:
2314:
2311:
2306:
2302:
2293:
2290:
2287:
2284:
2276:
2272:
2268:
2262:
2257:
2253:
2245:
2244:
2243:
2239:
2234:
2230:
2225:
2223:
2220:, called the
2217:
2213:
2209:
2198:
2196:
2195:Daniel Lazard
2192:
2187:
2165:
2160:
2156:
2150:
2146:
2140:
2136:
2132:
2129:
2126:
2121:
2117:
2111:
2107:
2103:
2100:
2097:
2094:
2089:
2085:
2081:
2078:
2075:
2070:
2066:
2060:
2056:
2052:
2049:
2046:
2041:
2037:
2033:
2030:
2025:
2021:
2017:
2012:
2008:
2004:
2001:
1999:
1989:
1985:
1979:
1975:
1969:
1965:
1961:
1958:
1953:
1949:
1945:
1942:
1939:
1936:
1931:
1927:
1921:
1917:
1913:
1910:
1907:
1902:
1898:
1894:
1889:
1885:
1881:
1878:
1873:
1869:
1863:
1859:
1855:
1852:
1849:
1846:
1841:
1837:
1833:
1830:
1827:
1825:
1815:
1811:
1805:
1801:
1797:
1794:
1789:
1785:
1781:
1778:
1773:
1769:
1763:
1759:
1755:
1752:
1749:
1744:
1740:
1736:
1731:
1727:
1723:
1720:
1717:
1714:
1711:
1706:
1702:
1698:
1695:
1690:
1686:
1682:
1679:
1674:
1670:
1664:
1660:
1656:
1653:
1651:
1644:
1630:
1629:
1628:
1607:
1604:
1601:
1598:
1595:
1590:
1586:
1582:
1579:
1574:
1570:
1566:
1563:
1560:
1558:
1548:
1544:
1540:
1537:
1532:
1528:
1524:
1521:
1518:
1515:
1512:
1509:
1504:
1500:
1496:
1493:
1488:
1484:
1478:
1474:
1470:
1467:
1462:
1458:
1452:
1448:
1444:
1441:
1438:
1433:
1429:
1425:
1422:
1417:
1413:
1409:
1407:
1394:
1390:
1386:
1383:
1380:
1377:
1374:
1371:
1366:
1362:
1358:
1355:
1350:
1346:
1342:
1339:
1336:
1333:
1328:
1324:
1320:
1314:
1311:
1303:
1299:
1295:
1292:
1289:
1286:
1278:
1274:
1270:
1265:
1261:
1253:
1250:
1239:
1238:
1237:
1231:
1221:
1217:
1188:
1184:
1180:
1173:
1169:
1165:
1162:
1159:
1154:
1150:
1146:
1143:
1140:
1137:
1132:
1128:
1122:
1118:
1114:
1111:
1108:
1105:
1100:
1096:
1092:
1089:
1086:
1081:
1077:
1073:
1067:
1065:
1060:
1048:
1044:
1040:
1033:
1029:
1025:
1022:
1019:
1014:
1010:
1006:
1003:
1000:
997:
994:
989:
985:
981:
978:
975:
970:
966:
962:
956:
954:
949:
937:
933:
929:
922:
918:
914:
911:
908:
905:
902:
899:
896:
893:
888:
884:
880:
874:
872:
867:
855:
851:
847:
840:
836:
832:
829:
826:
823:
820:
814:
812:
807:
796:
795:
794:
776:
773:
770:
767:
764:
761:
758:
753:
749:
745:
742:
737:
733:
729:
726:
721:
717:
709:
708:
707:
700:
691:
683:
679:
675:
656:
653:
650:
647:
644:
641:
638:
635:
630:
626:
622:
619:
614:
610:
606:
603:
598:
594:
590:
587:
582:
578:
574:
567:
566:
565:
562:
560:
559:Arthur Cayley
556:
555:Galois theory
552:
548:
543:
541:
535:
533:
514:
511:
508:
505:
502:
499:
494:
490:
482:
481:
480:
476:
472:
468:
464:
460:
456:
451:
441:
439:
434:
430:
426:
421:
417:
412:
408:
404:
400:
395:
393:
383:
381:
377:
373:
369:
365:
345:
342:
339:
336:
333:
330:
327:
322:
318:
314:
311:
306:
302:
298:
295:
290:
286:
282:
279:
274:
270:
266:
259:
258:
257:
256:of the form:
255:
249:
245:and assuming
242:
238:
232:
230:
226:
222:
221:local maximum
218:
213:
211:
207:
199:
195:
191:
187:
143:
140:
137:
134:
131:
128:
123:
119:
115:
112:
107:
103:
99:
96:
91:
87:
83:
80:
75:
71:
67:
64:
58:
52:
45:
44:
43:
41:
37:
33:
26:
21:
9648:Discriminant
9567:Multivariate
9530:
9417:(April 2024)
9409:
9376:
9354:the original
9338:
9321:
9314:
9306:
9302:
9298:
9277:
9259:
9252:
9248:
9243:
9239:
9235:
9227:
9223:
9204:
9195:
9191:
9173:
9169:
9163:
9154:
9150:
9144:
9135:
9131:
9125:
9104:
9092:
9076:"Trinomials
9069:
9062:Lazard (2004
9057:
9042:
9039:
9034:
9025:
9010:
9007:
9002:
8993:
8989:
8976:
8930:
8926:
8923:
8856:, given by:
8838:
8831:
8829:
8821:
8814:
8804:
8799:
8795:
8787:
8782:
8778:
8775:
8402:
8187:
8171:
8164:
8160:
8152:
8148:
8140:
8132:
8125:
8118:
8116:
7976:
7965:
7950:
7943:
7941:
7934:
7922:
7903:
7896:
7892:
7888:
7882:
7736:
7685:group theory
7675:by means of
7655:
7648:
7644:
7640:
7630:About 1835,
7629:
7616:
7610:
7605:
7277:
7136:
6963:
6105:
6101:
6089:
4842:
4481:
4430:
4427:
4409:
4402:
4400:
4291:
4223:
4129:
4126:
4120:
4114:
4108:
4102:
4095:
4091:
4085:
4075:
4065:
4059:
4053:
4047:
4040:
4033:
4029:
4023:
4016:
4012:
4009:
3875:
3872:
3754:
3509:is given by
3504:
3500:golden ratio
3472:
3459:
3451:
3443:
3435:
3427:
3420:
3416:
3413:
3269:
3265:
3261:
3257:
3253:
3249:
3199:
3116:
3030:
3009:
3000:
2989:Galois group
2986:
2803:
2799:
2795:
2784:
2777:
2767:
2753:
2743:
2735:
2726:
2723:
2712:
2703:
2694:
2691:
2593:
2432:
2426:
2420:
2417:
2237:
2226:
2215:
2211:
2207:
2204:
2188:
2184:
1626:
1225:
1219:
1215:
1212:
792:
698:
689:
681:
677:
671:
563:
551:group theory
544:
539:
536:
531:
529:
474:
470:
466:
462:
458:
454:
447:
435:
428:
424:
396:
390:Finding the
389:
367:
361:
253:
247:
240:
236:
233:
214:
194:real numbers
159:
42:of the form
35:
29:
9597:Homogeneous
9592:Square-free
9587:Irreducible
9452:Polynomials
9415:AMS Notices
9334:Ragni Piene
8929:= 1.5 x 10
8854:Hill sphere
7697:icosahedron
7693:Felix Klein
7659:. In 1858,
3022:(1 2 3 4 5)
3016:, of order
2233:irreducible
252:produces a
32:mathematics
9668:Categories
9557:Univariate
9185:References
9157:: 275–282.
9110:Klein 1888
7691:. Later,
4038:of degree
3873:where the
3043:, such as
2816:such that
2229:Carl Runge
452:, such as
225:derivative
206:polynomial
9674:Equations
9643:Resultant
9582:Trinomial
9562:Bivariate
9362:cite book
9114:Tóth 2002
8870:≈
8736:∓
8696:±
8598:∓
8523:±
8425:±
8216:π
8197:ω
8096:±
8081:ω
8039:±
8020:±
7564:−
7552:−
7540:−
7453:−
7444:−
7408:ℓ
7357:−
7316:ℓ
7250:−
7189:−
7151:ℓ
7080:−
7074:−
7068:−
7020:−
6994:−
6921:π
6888:∑
6818:−
6748:π
6715:∑
6663:−
6638:−
6622:−
6559:π
6526:∑
6498:−
6457:−
6394:π
6361:∑
6311:−
6295:−
6239:π
6223:
6165:−
6149:−
6041:−
5991:−
5915:−
5906:−
5890:−
5874:−
5836:−
5795:−
5757:−
5748:−
5732:−
5716:−
5506:−
5400:−
5368:−
5330:−
5305:−
5253:−
5209:−
5024:−
5001:−
4909:−
4900:−
4884:−
4868:−
4815:−
4789:−
4719:−
4675:−
4649:−
4605:−
4579:−
4531:−
4505:−
4355:ω
4345:−
4317:ω
4257:−
4135:de Moivre
3968:−
3945:−
3694:−
3685:−
3610:−
3589:−
3563:−
3520:−
3369:−
3309:−
3223:−
3217:∈
3214:β
3208:α
3173:−
3163:β
3148:β
3142:−
3136:−
3131:α
3090:−
3065:−
3059:−
3026:(1 2 4 3)
2858:−
2649:−
2635:±
2546:−
2521:−
2382:ν
2367:ν
2349:ν
2334:μ
2303:ν
2288:ν
2273:μ
2189:In 1888,
2130:−
2076:−
2050:−
1937:−
1879:−
1828:−
1696:−
1654:−
1642:Δ
1596:−
1538:−
1494:−
1468:−
1442:−
1410:−
1384:−
1356:−
1337:−
1312:−
1271:−
1141:−
1090:−
1023:−
979:−
897:−
830:−
506:−
500:−
450:reducible
403:quadratic
9628:Division
9577:Binomial
9572:Monomial
9336:(eds.).
8946:See also
7960:and the
7683:, using
4137:quintic
4063:. These
3470:, where
3423:+ 12 = 0
3272:− 21 = 0
1236:, where
1224:, named
397:Solving
364:radicals
234:Setting
40:function
9292:at the
9051:2369502
8135:is the
7883:to the
7632:Jerrard
6209:Roots:
6098:, with
3498:is the
3495:
3482:√
3477:
3464:√
3448:√
3432:√
2772:
2758:
2748:
2731:
2718:
2699:
1218:− 1024
703:
686:
465:+ 1 = (
431:+ 1 = 0
196:or the
9465:degree
9346:
9284:
9266:
9212:
9049:
8810:. The
8666:
8654:
8642:
8624:
8403:with:
8181:, and
8170:, and
7715:) and
7278:where
6879:Root:
6706:Root:
6691:
6687:
6517:Root:
6352:Root:
5189:Root:
5177:
5173:
4418:septic
4401:where
3457:, and
3425:. Let
3200:where
3037:nested
2692:where
2418:where
793:where
399:linear
212:five.
210:degree
200:, and
192:, the
160:where
9118:p. 66
9047:JSTOR
8986:(PDF)
8968:Notes
8938:and L
8848:and L
8179:Earth
5918:11200
4098:) = 0
4036:) = 0
4019:) = 0
2991:is a
473:+ 1)(
469:+ 1)(
407:cubic
392:roots
372:cubic
243:) = 0
186:field
38:is a
9454:and
9368:link
9344:ISBN
9282:ISBN
9264:ISBN
9210:ISBN
9174:XLVI
9136:XLVI
8820:and
8803:for
8793:and
8786:for
8143:the
8124:and
7973:SOHO
7971:and
7958:Gaia
7949:and
7923:See
7916:and
6104:= 10
5909:2000
5839:5888
5830:1000
5678:8320
5071:3125
4966:3750
4912:7412
4903:1505
4608:1000
3268:+ 55
3264:− 50
3260:+ 30
3024:and
2812:and
2789:and
2706:+ 3)
2424:and
2005:2250
1940:3750
1911:2000
1831:1600
1683:3125
1627:and
1599:1600
1564:4000
1181:3125
1074:3125
672:the
553:and
477:− 1)
409:and
374:and
180:and
34:, a
9463:By
9253:101
9015:doi
9011:151
8183:Sun
7975:at
7964:at
7899:= 0
7651:= 0
6220:cos
6108:+ 1
6094:th
6069:208
6060:170
6003:800
5994:360
5956:110
5893:600
5814:160
5760:656
5751:150
5669:800
5631:110
5333:400
5324:250
5068:979
5055:125
5052:462
4952:625
4818:300
4582:100
4043:– 1
3419:− 5
3256:− 5
2806:= 0
2785:If
2780:= 0
2715:+ 1
2702:5(4
2240:= 0
2218:= 0
2079:630
2034:108
1962:825
1882:900
1853:144
1798:108
1782:256
1724:560
1657:128
1583:320
1519:224
1471:176
1375:400
1359:240
1115:125
1093:625
1041:125
963:125
250:≠ 0
208:of
30:In
9670::
9413:,
9364:}}
9360:{{
9332:;
9305:+
9303:cx
9301:+
9276:,
9251:,
9247:,
9242:+
9240:ax
9238:+
9228:46
9226:,
9194:,
9172:.
9153:.
9134:.
9082:.
9043:10
8994:36
8992:.
8988:.
8185:.
8163:,
8147:,
8139:,
7901:.
7895:+
7891:−
7737:A
7723:.
7717:11
7699:,
7647:−
7643:+
6941:71
6927:23
6903:13
6853:25
6837:37
6821:28
6768:61
6754:21
6730:11
6666:13
6657:41
6641:17
6625:24
6573:41
6492:21
6460:16
6408:31
6314:21
6298:12
6243:11
6044:20
5978:20
5877:50
5798:40
5735:80
5719:20
5653:60
5591:13
5575:13
5572:10
5530:65
5527:29
5514:13
5469:17
5466:21
5453:17
5450:20
5405:13
5389:85
5308:20
5152:10
5143:30
5127:20
5111:20
5032:25
5029:11
5006:22
4887:20
4871:10
4792:25
4748:15
3480:1+
3475:=
3462:=
3446:=
3441:,
3430:=
3139:10
3062:10
3028:.
3018:20
2959:10
2914:11
2890:10
2802:+
2800:ax
2798:+
2756:=
2751:,
2729:=
2697:=
2646:20
2632:20
2579:0.
2549:11
2224:.
2214:+
2212:ax
2210:+
2104:16
2053:27
1756:16
1699:72
1541:64
1497:80
1445:16
1426:28
1340:16
1287:20
1144:25
1004:15
982:50
930:25
900:15
881:25
684:−
680:=
461:−
457:−
440:.
433:.
427:−
405:,
401:,
382:.
346:0.
231:.
176:,
172:,
168:,
164:,
9444:e
9437:t
9430:v
9370:)
9310:.
9307:d
9299:x
9244:b
9236:x
9218:.
9196:2
9155:I
9120:)
9108:(
9086:.
9017::
8940:2
8936:1
8931:m
8927:r
8906:3
8898:S
8894:M
8890:3
8884:E
8880:M
8873:R
8867:r
8850:2
8846:1
8842:S
8839:M
8835:E
8832:M
8825:1
8822:L
8818:2
8815:L
8808:1
8805:L
8800:m
8796:r
8791:2
8788:L
8783:m
8779:r
8772:.
8754:5
8750:R
8744:E
8740:M
8733:=
8730:f
8722:,
8717:4
8713:R
8709:2
8704:E
8700:M
8693:=
8690:e
8682:,
8679:)
8674:2
8670:L
8663:r
8660:o
8657:f
8651:0
8648:=
8645:d
8639:s
8636:u
8633:h
8630:t
8627:(
8619:3
8615:R
8611:)
8606:E
8602:M
8593:E
8589:M
8585:(
8582:+
8579:=
8576:d
8568:,
8563:2
8559:R
8555:3
8552:)
8547:E
8543:M
8539:+
8534:S
8530:M
8526:(
8520:=
8517:c
8509:,
8506:R
8503:3
8500:)
8495:E
8491:M
8487:+
8482:S
8478:M
8474:(
8471:+
8468:=
8465:b
8457:,
8454:)
8449:E
8445:M
8441:+
8436:S
8432:M
8428:(
8422:=
8419:a
8388:0
8385:=
8382:f
8379:+
8376:r
8373:e
8370:+
8365:2
8361:r
8357:d
8354:+
8349:3
8345:r
8341:c
8338:+
8333:4
8329:r
8325:b
8322:+
8317:5
8313:r
8309:a
8282:3
8278:R
8273:)
8268:E
8264:M
8260:+
8255:S
8251:M
8247:(
8244:G
8238:=
8231:2
8227:P
8220:2
8212:4
8206:=
8201:2
8174:S
8172:M
8167:E
8165:M
8161:m
8153:R
8149:r
8141:ω
8133:G
8129:1
8126:L
8122:2
8119:L
8102:)
8099:r
8093:R
8090:(
8085:2
8077:m
8074:=
8067:2
8063:r
8056:E
8052:M
8048:m
8045:G
8031:2
8027:)
8023:r
8017:R
8014:(
8007:S
8003:M
7999:m
7996:G
7980:1
7977:L
7969:2
7966:L
7954:1
7951:L
7947:2
7944:L
7897:t
7893:x
7889:x
7867:0
7864:=
7859:0
7855:a
7851:+
7848:x
7843:1
7839:a
7835:+
7830:2
7826:x
7820:2
7816:a
7812:+
7807:3
7803:x
7797:3
7793:a
7789:+
7784:4
7780:x
7774:4
7770:a
7766:+
7761:5
7757:x
7711:(
7709:7
7656:a
7649:a
7645:t
7641:t
7587:.
7582:]
7577:m
7572:2
7568:a
7561:)
7558:m
7555:4
7549:a
7546:(
7543:p
7537:)
7534:m
7531:4
7528:+
7525:a
7522:(
7519:b
7514:[
7507:2
7504:1
7498:=
7495:c
7470:,
7464:2
7460:m
7456:2
7450:p
7447:5
7441:)
7436:2
7432:a
7428:+
7423:2
7419:m
7415:4
7412:(
7405:=
7402:b
7377:,
7372:]
7365:2
7361:m
7354:)
7349:2
7345:a
7341:+
7336:2
7332:m
7328:4
7325:(
7320:2
7310:[
7303:4
7300:1
7294:=
7291:p
7263:0
7260:=
7257:c
7253:p
7246:)
7242:x
7238:b
7235:+
7230:2
7226:x
7221:a
7218:+
7213:3
7209:x
7204:2
7200:(
7196:p
7192:5
7184:5
7180:x
7157:m
7154:,
7148:,
7145:a
7122:0
7119:=
7116:a
7113:+
7110:x
7106:b
7103:+
7098:2
7094:x
7089:)
7086:b
7083:2
7077:1
7071:a
7063:2
7059:a
7055:(
7052:+
7047:3
7043:x
7038:)
7035:3
7032:+
7029:b
7026:+
7023:a
7017:(
7014:+
7009:4
7005:x
7000:)
6997:3
6991:a
6988:(
6985:+
6980:5
6976:x
6935:k
6931:)
6924:(
6918:i
6915:2
6909:e
6898:0
6895:=
6892:k
6862:1
6859:+
6856:x
6850:+
6845:2
6841:x
6834:+
6829:3
6825:x
6813:4
6809:x
6805:+
6800:5
6796:x
6762:k
6758:)
6751:(
6745:i
6742:2
6736:e
6725:0
6722:=
6719:k
6660:x
6654:+
6649:2
6645:x
6633:3
6629:x
6617:4
6613:x
6609:+
6604:5
6600:x
6567:k
6563:3
6556:i
6553:2
6547:e
6541:7
6536:0
6533:=
6530:k
6501:9
6495:x
6489:+
6484:2
6480:x
6476:5
6473:+
6468:3
6464:x
6452:4
6448:x
6444:+
6439:5
6435:x
6402:k
6398:6
6391:i
6388:2
6382:e
6376:5
6371:0
6368:=
6365:k
6336:5
6333:+
6330:x
6327:+
6322:2
6318:x
6306:3
6302:x
6290:4
6286:x
6282:+
6277:5
6273:x
6248:)
6236:k
6233:2
6227:(
6217:2
6193:1
6190:+
6187:x
6184:3
6181:+
6176:2
6172:x
6168:3
6160:3
6156:x
6152:4
6144:4
6140:x
6136:+
6131:5
6127:x
6106:k
6102:n
6092:n
6066:+
6063:x
6057:+
6052:3
6048:x
6036:5
6032:x
6006:)
6000:+
5997:x
5986:2
5982:x
5975:+
5970:3
5966:x
5962:5
5959:(
5953:+
5948:5
5944:x
5912:x
5901:2
5897:x
5885:3
5881:x
5869:5
5865:x
5833:x
5827:+
5822:2
5818:x
5811:+
5806:3
5802:x
5790:5
5786:x
5754:x
5743:2
5739:x
5727:3
5723:x
5711:5
5707:x
5681:)
5675:+
5672:x
5666:+
5661:2
5657:x
5650:+
5645:3
5641:x
5637:5
5634:(
5628:+
5623:5
5619:x
5588:3
5583:+
5580:x
5567:+
5562:5
5558:x
5522:+
5519:x
5511:4
5501:5
5497:x
5461:+
5458:x
5445:+
5440:5
5436:x
5408:2
5397:x
5392:8
5384:+
5379:3
5375:x
5371:5
5363:5
5359:x
5327:x
5321:+
5316:3
5312:x
5300:5
5296:x
5270:4
5263:5
5259:2
5248:3
5241:5
5237:2
5231:+
5226:2
5219:5
5215:2
5203:5
5199:2
5149:+
5146:x
5140:+
5135:2
5131:x
5124:+
5119:3
5115:x
5108:+
5103:5
5099:x
5063:+
5060:x
5047:+
5042:2
5038:x
5019:3
5015:x
5009:5
4996:5
4992:x
4963:+
4960:x
4955:4
4947:+
4942:5
4938:x
4906:x
4895:2
4891:x
4879:3
4875:x
4863:5
4859:x
4826:5
4822:s
4810:2
4806:x
4800:3
4796:s
4784:5
4780:x
4756:5
4752:s
4745:+
4740:2
4736:x
4730:3
4726:s
4722:5
4714:5
4710:x
4686:5
4682:s
4678:3
4670:2
4666:x
4660:3
4656:s
4652:5
4644:5
4640:x
4616:5
4612:s
4600:2
4596:x
4590:3
4586:s
4574:5
4570:x
4546:5
4541:5
4537:s
4526:2
4522:x
4516:3
4512:s
4508:2
4500:5
4496:x
4482:s
4468:b
4465:+
4460:2
4456:x
4452:a
4449:+
4444:5
4440:x
4410:ω
4405:i
4403:y
4386:,
4377:5
4371:i
4367:y
4359:k
4350:a
4339:5
4333:i
4329:y
4321:k
4313:=
4308:k
4304:x
4277:,
4273:0
4270:=
4265:5
4261:a
4254:y
4251:b
4248:+
4243:2
4239:y
4209:,
4205:0
4202:=
4199:b
4196:+
4193:x
4188:2
4184:a
4180:5
4177:+
4172:3
4168:x
4164:a
4161:5
4158:+
4153:5
4149:x
4130:Q
4121:p
4115:p
4109:p
4103:p
4096:x
4094:(
4092:P
4086:Q
4076:p
4066:p
4060:Q
4054:p
4048:P
4041:p
4034:y
4032:(
4030:Q
4024:p
4017:x
4015:(
4013:P
3995:.
3991:0
3988:=
3981:5
3977:5
3973:1
3965:y
3958:3
3954:5
3950:8
3940:2
3936:y
3930:5
3927:4
3922:+
3917:3
3913:y
3909:4
3906:+
3901:4
3897:y
3878:i
3876:y
3858:,
3851:5
3845:4
3841:y
3835:+
3829:5
3823:3
3819:y
3813:+
3807:5
3801:2
3797:y
3791:+
3785:5
3779:1
3775:y
3769:=
3766:x
3740:,
3733:5
3728:)
3725:c
3722:+
3719:b
3716:(
3711:2
3707:)
3703:c
3700:+
3697:a
3691:(
3679:5
3672:2
3668:)
3664:c
3661:+
3658:b
3655:(
3652:)
3649:c
3646:+
3643:a
3640:(
3634:+
3628:5
3621:2
3617:)
3613:c
3607:b
3604:(
3601:)
3598:c
3595:+
3592:a
3586:(
3580:+
3574:5
3569:)
3566:c
3560:b
3557:(
3552:2
3548:)
3544:c
3541:+
3538:a
3535:(
3529:=
3526:x
3523:c
3505:x
3492:2
3489:/
3484:5
3473:φ
3466:5
3460:c
3452:φ
3450:2
3444:b
3436:φ
3434:2
3428:a
3421:x
3417:x
3399:.
3394:4
3389:)
3383:5
3379:2
3374:(
3364:3
3359:)
3353:5
3349:2
3344:(
3339:+
3334:2
3329:)
3323:5
3319:2
3314:(
3303:5
3299:2
3294:+
3291:1
3288:=
3285:x
3270:x
3266:x
3262:x
3258:x
3254:x
3235:}
3232:1
3229:,
3226:1
3220:{
3211:,
3185:,
3180:4
3176:1
3168:5
3160:+
3153:5
3145:2
3102:.
3097:4
3093:1
3085:5
3080:+
3073:5
3068:2
3013:5
3010:F
3004:5
3001:S
2967:.
2955:l
2951:+
2946:2
2942:m
2936:)
2933:m
2930:2
2927:+
2922:5
2918:l
2911:(
2908:4
2902:=
2899:b
2886:l
2882:+
2877:2
2873:m
2867:)
2864:m
2861:4
2853:5
2849:l
2845:3
2842:(
2839:l
2836:5
2830:=
2827:a
2814:m
2810:l
2804:b
2796:x
2791:b
2787:a
2778:a
2768:l
2764:/
2761:1
2754:e
2744:l
2740:/
2736:m
2734:−
2727:c
2713:ν
2709:/
2704:ν
2695:a
2676:)
2670:)
2667:a
2664:+
2661:5
2658:(
2655:)
2652:a
2643:(
2638:2
2629:+
2626:a
2622:(
2616:5
2613:4
2608:=
2605:b
2576:=
2570:1
2567:+
2562:2
2558:c
2552:)
2543:c
2540:2
2537:(
2532:5
2528:e
2524:4
2515:+
2512:x
2506:1
2503:+
2498:2
2494:c
2488:)
2485:3
2482:+
2479:c
2476:4
2473:(
2468:4
2464:e
2460:5
2454:+
2449:5
2445:x
2427:ν
2421:μ
2403:0
2400:=
2394:1
2391:+
2386:2
2376:)
2373:3
2370:+
2364:4
2361:(
2358:)
2355:1
2352:+
2346:2
2343:(
2338:5
2330:4
2324:+
2321:x
2315:1
2312:+
2307:2
2297:)
2294:3
2291:+
2285:4
2282:(
2277:4
2269:5
2263:+
2258:5
2254:x
2238:a
2216:b
2208:x
2166:.
2161:2
2157:r
2151:2
2147:q
2141:3
2137:p
2133:4
2127:s
2122:3
2118:q
2112:3
2108:p
2101:+
2098:s
2095:r
2090:3
2086:q
2082:p
2071:2
2067:r
2061:4
2057:q
2047:s
2042:5
2038:q
2031:+
2026:2
2022:s
2018:r
2013:2
2009:q
2002:+
1990:2
1986:s
1980:2
1976:q
1970:2
1966:p
1959:+
1954:3
1950:s
1946:q
1943:p
1932:2
1928:s
1922:2
1918:r
1914:p
1908:+
1903:2
1899:s
1895:r
1890:3
1886:p
1874:3
1870:r
1864:2
1860:q
1856:p
1850:+
1847:s
1842:3
1838:r
1834:q
1816:2
1812:s
1806:5
1802:p
1795:+
1790:5
1786:r
1779:+
1774:3
1770:r
1764:4
1760:p
1753:+
1750:s
1745:2
1741:r
1737:q
1732:2
1728:p
1721:+
1718:s
1715:r
1712:q
1707:4
1703:p
1691:4
1687:s
1680:+
1675:4
1671:r
1665:2
1661:p
1645:=
1608:q
1605:s
1602:r
1591:3
1587:r
1580:+
1575:2
1571:s
1567:p
1561:+
1549:4
1545:q
1533:2
1529:q
1525:r
1522:p
1516:+
1513:q
1510:s
1505:2
1501:p
1489:2
1485:r
1479:2
1475:p
1463:2
1459:q
1453:3
1449:p
1439:r
1434:4
1430:p
1423:+
1418:6
1414:p
1400:)
1395:4
1391:p
1387:3
1381:q
1378:s
1372:+
1367:2
1363:r
1351:2
1347:q
1343:p
1334:r
1329:2
1325:p
1321:8
1318:(
1315:z
1309:)
1304:2
1300:p
1296:3
1293:+
1290:r
1284:(
1279:2
1275:z
1266:3
1262:z
1254:=
1251:P
1234:z
1222:Δ
1220:z
1216:P
1189:5
1185:a
1174:5
1170:b
1166:4
1163:+
1160:c
1155:3
1151:b
1147:a
1138:d
1133:2
1129:b
1123:2
1119:a
1112:+
1109:e
1106:b
1101:3
1097:a
1087:f
1082:4
1078:a
1068:=
1061:s
1049:4
1045:a
1034:4
1030:b
1026:3
1020:c
1015:2
1011:b
1007:a
1001:+
998:d
995:b
990:2
986:a
976:e
971:3
967:a
957:=
950:r
938:3
934:a
923:3
919:b
915:4
912:+
909:c
906:b
903:a
894:d
889:2
885:a
875:=
868:q
856:2
852:a
848:5
841:2
837:b
833:2
827:c
824:a
821:5
815:=
808:p
789:,
777:0
774:=
771:s
768:+
765:y
762:r
759:+
754:2
750:y
746:q
743:+
738:3
734:y
730:p
727:+
722:5
718:y
699:a
697:5
694:/
690:b
682:y
678:x
657:,
654:0
651:=
648:f
645:+
642:x
639:e
636:+
631:2
627:x
623:d
620:+
615:3
611:x
607:c
604:+
599:4
595:x
591:b
588:+
583:5
579:x
575:a
532:r
515:0
512:=
509:r
503:x
495:5
491:x
475:x
471:x
467:x
463:x
459:x
455:x
429:x
425:x
368:n
366:(
343:=
340:f
337:+
334:x
331:e
328:+
323:2
319:x
315:d
312:+
307:3
303:x
299:c
296:+
291:4
287:x
283:b
280:+
275:5
271:x
267:a
248:a
241:x
239:(
237:g
202:a
182:f
178:e
174:d
170:c
166:b
162:a
144:,
141:f
138:+
135:x
132:e
129:+
124:2
120:x
116:d
113:+
108:3
104:x
100:c
97:+
92:4
88:x
84:b
81:+
76:5
72:x
68:a
65:=
62:)
59:x
56:(
53:g
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