Quintic and Higher Degree Polynomial Equations

[Pikkugnome]

TL;DR Summary

Formula for roots of higher degree polynomials.

What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?

 

[fresh_42]

TL;DR Summary: Formula for roots of higher degree polynomials.

What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?

There are none for the general case, meaning, it has been proven that there cannot be such solutions.

I doubt that the class of polynomials of a certain degree higher than four that do have radical solutions can be described in a way that allows the listing of their roots. You can construct such polynomials by multiplying terms ##x-x_k ## but that won't exhaust these classes.

 

[FactChecker]

Only certain quintic polynomials are solvable. See this. It is related to Galois theory, which I know nothing about. I think that the higher degree polynomials are even more complicated. People usually resort to numeric algorithms for fourth degree and higher polynomials.

 

[fresh_42]

I have tried to shed some light on Galois theory and the solvability in this article
https://www.physicsforums.com/insights/intro-to-evariste-galois-and-galois-theory/

van der Waerden's book about abstract algebra is available in English and contains a detailed treatise of Galois theory.