An Overview of the True Quintic Formula...and Why You Should Never Use It

Many people know that the general quintic equation is unsolvable in terms of using finite elementary operations from Abel-Ruffini theorem, however, you can find a solution to it by using polynomial transformations and different non-elementary means. This video outlines one possible solution method using power series and hypergeometric functions. However, essentially all of this is a proof of concept since this solution, along with many others, is simply way too long and complicated to be used in practice, especially in back-substituted form, and it even suffers computationally with symbolic coefficients taking up so much space.

Timecodes:
0:00 - Intro and transformation
(Let principal's y² coefficient be t instead of u**)
11:40 - Bring-Jerrard principal root
20:45 - Bring-Jerrard remaining roots
(Must have u ≠ 0, could also divide out principal soln. and solve the remaining quartic**)
30:08 - Back-substitution and closer
(Elliptic soln. has actually been written back-subbed for the
original quintic, though still not with the original coefficients**)

Sources and other tidbits:
The quintic equation: https://mathworld.wolfram.com/Quintic...
Check if a quintic is solvable in radicals: https://math.stackexchange.com/a/1904...
Tschirnhaus transformation: https://en.wikipedia.org/wiki/Tschirn...
(Note that this transformation can also be rational, not just polynomial**)
Resultant of two polynomials: https://en.wikipedia.org/wiki/Resultant
(Also see Newton's Power Sum Identities for an alternate way to get coeffs**)
Transforming a quintic guide: https://math.stackexchange.com/a/542228
Lagrange inversion/reversion theorem: https://mathworld.wolfram.com/Lagrang...
Bring radical: https://en.wikipedia.org/wiki/Bring_r...
Generalized hypergeometric function: https://en.wikipedia.org/wiki/General...
Alternate differential resolvent solution: https://math.stackexchange.com/a/3789323
Alternate elliptic function solution: https://math.stackexchange.com/a/541027
Overall quintic solution summary: https://arxiv.org/pdf/math/0005026.pdf
Wolfram Mathematica guide to solving the quintic: https://web.archive.org/web/201407010...
(If you're insane, modular functions can solve any polynomial equation: https://en.wikipedia.org/wiki/Thomae%...)

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