On Aug 21, 7:03 am, w...@itd.nrl.navy.mil (J. B. Wood) wrote:
> In article <1187519178.733837.194...@19g2000hsx.googlegroups.com>, Ian
>
> Parker <ianpark...@gmail.com> wrote:
> > On 19 Aug, 08:39, "Jon G." <jon8...@peoplepc.com> wrote:
> > >http://mypeoplepc.com/members/jon8338/polynomial/
>
> > Roots can only be found explicitly (as opposed to Newton's
> > approximation)if there is a splitting field. The cubic and quartic can
> > be decomposed into quadratics. Golois proved, and his proof is now
> > universally accepted, that a general quintic and higer power is not
> > analytically soluble.
>
> >   - Ian Parker
>
> Thanks, Ian, but I think I alluded to that in my OP.  College-level
> algebra texts usually have the solutions for the cubic and quartic
> (biquadratic).  What is being asked is whether closed-form non-algebraic
> solutions exist for the roots of polynomials of quintic and higher
> degree.  Just so there's no confusion, and not intending to insult
> anyone's intelligence, by "closed-form" I mean variables (in this case
> representing the roots) to the left of the equals sign and some valid
> mathematical operation(s) on the polynomial coefficients to the right of
> the equals sign.  Sincerely,
>
> John Wood (Code 5550)        e-mail: w...@itd.nrl.navy.mil
> Naval Research Laboratory
> 4555 Overlook Avenue, SW
> Washington, DC 20375-5337

There are generalizations of the result
for quintics that you allude to.

First, expressions for polynomial roots
can be given in terms of Bring radicals
(or ultraradicals, as they are also known).

http://en.wikipedia.org/wiki/Bring_radical

Second, the Bring radical values can be
expressed in terms of elliptic modular
functions (Hermite, 1858) or generalized
hypergeometric functions (M. L. Glassner,
1994).  See arXiv.org paper here:

http://xxx.lanl.gov/abs/math.CA/9411224

regards, chip