Quintic and Higher Degree Polynomial Equations

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Discussion Overview

The discussion centers on the existence of root formulas for quintic and higher degree polynomial equations, particularly in relation to solutions expressed in radicals. It explores theoretical aspects, including connections to Galois theory and the implications for solvability.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants assert that there are no general root formulas for fifth and higher degree polynomials that yield solutions in radicals, citing proven results regarding their solvability.
  • Others propose that while certain quintic polynomials may be solvable, the general case remains complex and is often approached through numerical algorithms for degrees four and higher.
  • A participant mentions Galois theory as a relevant framework for understanding the solvability of these polynomials, although they express a lack of familiarity with the theory itself.
  • One participant references an article aimed at clarifying Galois theory and its implications for polynomial solvability, suggesting further reading for those interested.

Areas of Agreement / Disagreement

Participants generally agree that there are complexities surrounding the solvability of quintic and higher degree polynomials, but multiple competing views remain regarding the existence of radical solutions and the applicability of Galois theory.

Contextual Notes

The discussion does not resolve the limitations of existing theories or the specific conditions under which certain polynomials may be solvable. There is also a lack of consensus on the extent to which Galois theory can be applied to these problems.

Pikkugnome
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TL;DR
Formula for roots of higher degree polynomials.
What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?
 
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Pikkugnome said:
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.
 
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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.
 
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