Quintic and Higher Degree Polynomial Equations

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SUMMARY

There are no general root formulas for quintic and higher degree polynomial equations that yield solutions in radicals, as proven by Galois theory. Only specific quintic polynomials can be solved using radicals, while the general case remains unsolvable. For polynomials of degree four and higher, numeric algorithms are typically employed for root finding. Resources such as van der Waerden's book on abstract algebra provide in-depth insights into Galois theory and its implications on polynomial solvability.

PREREQUISITES
  • Understanding of polynomial equations and their degrees
  • Familiarity with Galois theory and its principles
  • Knowledge of numeric algorithms for root finding
  • Basic concepts of abstract algebra
NEXT STEPS
  • Study Galois theory in detail to understand polynomial solvability
  • Explore numeric algorithms for root finding in polynomials
  • Read van der Waerden's book on abstract algebra for advanced concepts
  • Investigate specific quintic polynomials that can be solved using radicals
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the complexities of polynomial equations and their solvability.

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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