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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.
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    $\begingroup$ If you interested in the modern continuation, Umemura extended the solution to all degrees in terms of Siegel's modular function in 1984. He also gives some insight on the idea and historical remarks for the quintic involving Hermite, Kronecker, Klein and Jordan, with a connection to Jacobi at the end. For more background see King's book Beyond the Quartic Equation, who also mentions Gordan and Kiepert. $\endgroup$ – Conifold Mar 21 at 0:04
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In general, transformation of elliptic integrals (or differentials) is finding algebraic solutions $F(x,y)=0$ of a differential equation $$\frac{dx}{\sqrt{f(x)}}=\frac{dy}{\sqrt{g(y)}},$$ where $f,g$ are polynomials of degree 3.

First such transformation was discovered by Landen in 1775, and it is called Landen's transformation. Independently it was discovered by Gauss 1790 when he studied the arithmetic-geometric mean (previously studied by Lagrange in 1785). But as usual, Gauss gets all credit for anything he touched.

The theory of transformations leads to certain algebraic equations called classical modular equations which Hermite used to solve the quintic.

You can read the details in Klein's book Lectures on Icosahedron and solution of 5-th degree equation. For a modern exposition, you can look to the book of Jonathan and Peter Borwein, Pi and AGM.

Original papers: Hermite C. R. 46 (1858) 508-515. Kronecker (a simplified proof): C. R. 46 (1858) 1150-1152.

Generalization to equations of arbitrary degree: H. Umemura, Solving algebraic equations with theta-constants, Appendix I to the book of D. Mumford, Tata lectures on Theta, 1983.

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  • $\begingroup$ your answer does help (that's why i voted your answer), but only a little. Your answer helped me understand that the "transformation problem" means to rewrite the differential under the sign integral by substituting the integrand variable with a new variable that is connected to the original variable through algebraic relation $f(x,y)$. But can you broad the discussion about the connection to the quintic equation? and what is the intuitive meaning of transformation of an elliptic integral - can it be explained in a more geometrical manner? $\endgroup$ – user2554 Mar 21 at 15:14
  • $\begingroup$ Long discussions are not encouraged in the comments. Why don't you look at the literature I mentioned, or to the original papers? They are not long. I added the references. $\endgroup$ – Alexandre Eremenko Mar 21 at 23:23
  • $\begingroup$ Is this transformation also studied by Euler when he extended Fagnano's result on doubling of the lemniscate arc? $\endgroup$ – timur Aug 6 at 15:21

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