Has it been proven that quintic equations cannot be solved by *any* formula?

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Discussion Overview

The discussion revolves around the solvability of quintic equations, particularly in relation to Abel's theorem and Galois theory. Participants explore whether quintic equations can be solved using methods beyond traditional arithmetic and root operations, including the potential use of infinite series and other functions.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Historical

Main Points Raised

  • Some participants reference Abel's theorem, asserting that quintic equations cannot be solved by arithmetic and root operations, but question if other functions could be utilized.
  • One participant states that Abel's theorem has proven the general quintic equation cannot be solved in the traditional sense.
  • Another participant suggests that while finite expressions cannot solve quintics, infinite series and hypergeometric functions might provide solutions.
  • The Bring radical is mentioned as a potential method, though it is acknowledged to involve infinite series.
  • Galois theory is cited as a framework proving that quintic equations cannot be expressed using basic operations and roots, yet other methods may still exist.
  • Historical context is provided regarding mathematicians' efforts to solve polynomial equations, highlighting the contributions of Tartaglia and Ferrari, and the eventual proof by Abel.
  • Participants discuss the relevance of the Galois group of the roots, noting that the solvability of the group affects the ability to solve the polynomial.
  • There is mention that roots can also be represented by infinite series, adding another layer to the discussion of solvability.

Areas of Agreement / Disagreement

Participants express differing views on the solvability of quintic equations, with some asserting that traditional methods are insufficient while others propose alternative approaches. The discussion remains unresolved regarding the applicability of these alternative methods.

Contextual Notes

Participants acknowledge limitations in their arguments, particularly concerning the definitions of solvability and the nature of the functions being considered. The discussion also reflects varying interpretations of Abel's theorem and Galois theory.

swampwiz
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It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?
 
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swampwiz said:
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?

If you restrict yourself to finite expressions, yes, that is true. However, if you allow such things as infinite series and the like, you can solve quintic equations---in terms of hypergeometric functions. See, eg.,
http://mathworld.wolfram.com/QuinticEquation.html
 
OK, so it looks like the Bring radical could work, but it's an infinite series.
 
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swampwiz said:
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?

Through Galois theory, it is proven that the general solution of a polynomial of degree at least 5 is not expressible in terms of the operations addition, multiplication (and their inverses) and taking roots.

Other ways are still possible.
 
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Historically, mathematicians would compete for solving various types of polynomials. Tartaglia, an amateur Italian mathematician came up with the general formula for roots of a cubic and created a poem encoding the formula to prevent others from claiming they found it first.

https://www.storyofmathematics.com/16th_tartaglia.html

From there Ferrari, another younger mathematician conquered the quartics and the quintics remained unsolvable until Abel definitively proved they were by Galois theory.
 
It depends on whether the associated Galois group of the roots is a solvable group.
 
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And we know both that, of course there are nonsolvable groups and that these are the Galois groups of some polynomials.
 
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of course "roots" are also given by infinite series.
 

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