The general quintic equation cannot be solved by radicals and is shown in a landmark and far reaching work of Galois from 1832, which became a template of modern Group theory and Galois theory. However, the general quintic equation can be reduced to a Bring radical form (using the Tschirnhaus transformation), and this form of the quintic can somehow be solved using ideas from elliptic functions theory. I'm really not familiar with these materials, and that is why I am asking this question.
According to what I have read, Hermite based his construction on results by Jacobi and remarks made by Galois himself (in Galois' last paper) - these results are concerned with the so-called "transformation problem of elliptic integrals"; an $n$th order transformation of elliptic integral leads to a modular equation which is actually an $(n+1)$th degree polynomial equation in two variables, and these variables are somehow connected to the elliptic integral. Hermite based his proof on the $5$th order transformation of Jacobi.
For the completeness of historical discussion, according to p.4 of the book "Hessian Polyhedra, Invariant Theory And Appell Hypergeometric Functions", transformations of order 3,5 and 7 had been known to Gauss since 1808, and according to another source Gauss also made some very significant remarks on the problem of transformation of any odd order.
My questions are therefore both explanatory and historical:
- What is the "transformation problem of elliptic integrals" and how is it related to the solution of polynomial equations? I want to understand a bit more about the meaning of Hermite's achievement.
- I just want to know if anyone can help locate these transformations in Gauss' Nachlass. I think it's somewhere in the section on elliptic functions in volume 3 of his work, but I'm completely unable to recognize familiar patterns in the flood of formulae in those writings.