Editing Quintic function (section)

==Beyond radicals==

About 1835, [[George Jerrard|Jerrard]] demonstrated that quintics can be solved by using [[ultraradical]]s (also known as Bring radicals), the unique real root of {{math|''t''<sup>5</sup> + ''t'' − ''a'' {{=}} 0}} for real numbers {{math|''a''}}. In 1858, [[Charles Hermite]] showed that the Bring radical could be characterized in terms of the Jacobi [[theta function]]s and their associated [[elliptic modular function]]s, using an approach similar to the more familiar approach of solving [[cubic equation]]s by means of [[trigonometric function]]s.  At around the same time, [[Leopold Kronecker]], using [[group theory]], developed a simpler way of deriving Hermite's result, as had [[Francesco Brioschi]].  Later, [[Felix Klein]] came up with a method that relates the symmetries of the [[icosahedron]], [[Galois theory]], 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 [[generalized hypergeometric function]]s.<ref>{{Harv|Klein|1888}}; a modern exposition is given in {{Harv|Tóth|2002|loc=Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, [https://books.google.com/books?id=i76mmyvDHYUC&pg=PA66 p. 66]}}</ref> Similar phenomena occur in degree {{math|7}} ([[septic equation]]s) and {{math|11}}, as studied by Klein and discussed in {{slink|Icosahedral symmetry|Related geometries}}.

===Solving with Bring radicals===
{{Main article|Bring radical}}

A [[Tschirnhaus transformation]], which may be computed by solving a [[quartic equation]], reduces the general quintic equation of the form 
:<math>x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\,</math> 
to the [[Bring–Jerrard normal form]] {{math|''x''<sup>5</sup> − ''x'' + ''t'' {{=}} 0}}.

The roots of this equation cannot be expressed by radicals. However, in 1858, [[Charles Hermite]] published the first known solution of this equation in terms of [[elliptic function]]s.<ref name="hermite">{{cite journal
 | last = Hermite
 | first = Charles
 | year = 1858
 | title =  Sur la résolution de l'équation du cinquième degré 
 | journal = Comptes Rendus de l'Académie des Sciences
 | volume = XLVI
 | issue = I
 | pages = 508–515}}</ref>
At around the same time [[Francesco Brioschi]]<ref>
{{cite journal
 | last = Brioschi
 | first = Francesco
 | year = 1858
 | title =  Sul Metodo di Kronecker per la Risoluzione delle Equazioni di Quinto Grado 
 | journal = Atti Dell'i. R. Istituto Lombardo di Scienze, Lettere ed Arti
 | volume = I
 | pages = 275–282}}</ref> 
and [[Leopold Kronecker]]<ref>
{{cite journal
 | last = Kronecker
 | first = Leopold
 | year = 1858
 | title =  Sur la résolution de l'equation du cinquième degré, extrait d'une lettre adressée à M. Hermite 
 | journal = Comptes Rendus de l'Académie des Sciences
 | volume = XLVI
 | issue = I
 | pages = 1150–1152}}</ref>
came upon equivalent solutions.

See [[Bring radical]] for details on these solutions and some related ones.