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From: Frederick Williams on 17 Jun 2010 14:38
achille wrote:
>
> On Jun 17, 1:15 am, "F/32 Eurydice" <f32euryd...(a)sbcglobal.net> wrote:
> > On Jun 16, 12:20 pm, Frederick Williams
> >
> > <frederick.willia...(a)tesco.net> wrote:
> > > F/32 Eurydice wrote:
> >
> > > > What's the highest degree polynomial for which all roots can be
> > > > written in terms of known functions? TIA.
> >
> > > I think that roots can be written as functions of coefficients for all
> > > degrees: theta functions do for the lot.

> >
> > Can you give me a convenient reference for this?
>
> In 1870, Jordan first showed any algebraic equation can be
> solved using modular functions[1]. In 1984, Umemura Hiroshi
> has written down an explicit basis for expressing the roots
> on any algebraic equation by higher genus theta functions[2].
>
> [1] C.Jordan - Trait'e des Substitutions et des E'quations
> Alge'briques, Gautheirs-Villars, Paris 1870.
> [2] H.Umemura - Resolution of Algebraic equations by Theta Constants.
> appear as an appendix in the book "Tata Lectures on Theta II:
> Jacobian theta functions and differential equations",
> D. Mumford, ed. pp, 3.261-272, Birkhauser, Boston, 1984.
>
> BTW, I copied these reference from R. Bruce King's book
> "Beyond the Quartic equation (Modern Birkhauser Clasics)".
> It is a book for non-specialist which discuss how to solve
> quintic polynomials and beyond.

Thank you, achille. It was King's book that I would have referred to
had I returned to the thread soon enough. I know nothing about the
matter beyond what's in King's book.

--
I can't go on, I'll go on.
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