For example, using Bring radicals or elliptic functions to solve quintic equations. Wikipedia says that similar methods can be used for higher degree polynomials, but I'm struggling on finding resources on the general theory of using special functions in addition to radicals to solve polynomial equations. I tried looking up "hyperradicals", but this name seems to be used for other purposes. Is there a name for this general theory? Has the theory I am speaking of even been developed much at all?

  • $\begingroup$ This would probably be more complicated than just calculating the roots numerically. $\endgroup$ – Peter Jan 7 '16 at 21:54
  • $\begingroup$ Of course, and that is probably true in the quintic case as well, I am merely interested in the concept, and the fact that things like Bring radicals and elliptic functions have been used to solve quintics indicates to me that others have as well, so I was wondering if there was some general theory of these sorts of things. $\endgroup$ – Nathan BeDell Jan 7 '16 at 21:58
  • $\begingroup$ Are there at least criterions when a quintic can be solved with the help of the additional methods ? $\endgroup$ – Peter Jan 7 '16 at 22:00
  • $\begingroup$ The general quintic is always solvable using the bring radical, see this. $\endgroup$ – Nathan BeDell Jan 7 '16 at 22:05
  • $\begingroup$ OK, and for the sixtic ? $\endgroup$ – Peter Jan 7 '16 at 22:08

The theory of elliptic functions in general is sufficient to solve polynomials of any order, as is shown in Jordan's 1870 Treatise on Substitutions and Algebraic Equations. This is discussed here, for example.

  • $\begingroup$ Please give name, as the link is dead. $\endgroup$ – jiten Dec 22 '17 at 2:20

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