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From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations.

How does it imply that we can't have a formula that involves some other mathematics thus allowing us to find all the roots of any polynomial? Do abstract algebra and Galois theory have a role in this I'm not aware of?

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    $\begingroup$ It does not, actually. We can construct solutions of quintics using mathematical function, yes. Lame way to do this : $f(a)$ where $f(x)$ is defined to be the inverse function of $x^5 + x$. This is the "so-called" bring radical. Smart way to do this : Elliptic functions. Explicitly given here. $\endgroup$ Commented May 9, 2014 at 19:48

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It doesn't imply there isn't some other formula. Hermite, for example, found a solution for the general quintic using theta functions.

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    $\begingroup$ Could you explain better? $\endgroup$
    – seldon
    Commented Apr 29, 2014 at 17:34
  • $\begingroup$ Not much, I'm afraid. I don't actually know how it was done, only that it was. Wikipedia has something here: en.wikipedia.org/wiki/Bring_radical. There are some sources linked in this discussion: math.stackexchange.com/questions/32616/… $\endgroup$
    – jdc
    Commented Apr 29, 2014 at 17:37
  • $\begingroup$ Probably It would be too complex for me to understand it now. Thank you! $\endgroup$
    – seldon
    Commented Apr 29, 2014 at 17:57

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