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jedishrfu

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https://en.m.wikipedia.org/wiki/Abel–Ruffini_theorem

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Ray Vickson

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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

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TeethWhitener

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Of some relevance:

https://en.m.wikipedia.org/wiki/Bring_radical

https://en.m.wikipedia.org/wiki/Bring_radical

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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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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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lavinia

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https://www.maa.org/press/periodica...nomials-with-simple-curves-sextic-and-quintic

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jedishrfu

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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.

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WWGD

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

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WWGD

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mathwonk

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

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