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

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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?
 

Ray Vickson

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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?
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.,
 
OK, so it looks like the Bring radical could work, but it's an infinite series.
 
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Math_QED

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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?
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.
 

WWGD

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It depends on whether the associated Galois group of the roots is a solvable group.
 

WWGD

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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.
 

mathwonk

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of course "roots" are also given by infinite series.
 

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