This answer and this wikipedia section describe solving the quintic using The Hermite–Kronecker–Brioschi method. The functions involved seem a bit overpowered, but it is a relatively simple method.

I cannot read it's proof, since it is in french, but I'm not really looking for a proof (although one will probably help). I'm just looking for a high-level explanation of how the roots of quintics relate to hypergeometric functions, elliptic integrals, jacobi theta functions, modular forms, etc.

It doesn't seem intuitive at all, but you can plug in the numbers, and it works. How? Why?

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    $\begingroup$ Well, it works because they draw those methods straight through algebra. Lots of it. And some other minor pieces of math-o-magic $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 12:10
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    $\begingroup$ Say, what's the question? You say it doesn't seem intuitive, so what do you want? A proof? In English? $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 12:13
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    $\begingroup$ Perhaps you should see here on how to reduce it to the Bring-Jerrard form? $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 12:16
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    $\begingroup$ Well, I can't help but wonder how far this will manage to get. $\endgroup$ – Simply Beautiful Art Jan 3 '17 at 12:20
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    $\begingroup$ The basic idea is that the roots of a quintic in Bring-Jerrard form can be expressed as rational functions of the roots of a sextic equation. On the other hand it is known that the modular equation of degree $5$ is a sextic equation whose roots can be expressed in terms of theta functions. Now it is a matter of some smart algebraic computation (for example done by Hermite) to link the sextic modular equation with the corresponding quintic in Bring-Jerrard form. See full details with proofs in Elliptic Functions by Armitage and Eberlein. $\endgroup$ – Paramanand Singh Jan 4 '17 at 9:19

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