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I was looking into the Abel-Ruffini theorem stating that the roots of high order polynomials cannot have solutions written as a (finite??) sum of radicals. My question is whether there exist infinite summation formula for finding the roots*.

I'm not well versed in Galois theory, but looking at the proof, the theorem seems to rely on finite manipulations of the coefficients and variables defining the polynomial (over some field). Am I naive in wondering if one can utilize continuous transformations of the coefficients to find an infinite summation formula?

Edit: *General Formula for roots not individual roots.

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    $\begingroup$ That of course makes the situation more complex as we will need some topology $\endgroup$ Commented Jul 10, 2021 at 17:57
  • $\begingroup$ Just in case you're not aware: the roots of some polynomials of degree $> 4$ can be expressed in finitely many steps (in terms of rational numbers and the operations $+, -, \cdot, /$, and $\sqrt[n]{\:}$); a simple example is $x^5 - 2$. Abel-Ruffini only tells us that there are other polynomials (e.g. $x^5 - x - 1$) whose roots can't be expressed in this way, and in particular there's no general formula of this type that works simultaneously for all polynomials over $\mathbb Q$ with any given degree $d > 4$. $\endgroup$ Commented Jul 10, 2021 at 18:40
  • $\begingroup$ @RaviFernando Yes sorry my wording wasn't correct in the question. $\endgroup$
    – iglizworks
    Commented Jul 10, 2021 at 18:58

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That is not naive at all. Actually, several XIXth mathematicians (Hermite, Kronecker, Brioshi, and Klein) were able to solve the general quintic equation using more complex tools than arithmetic operations and radicals. For instance, it is possible to solve every quintic equation using generalized hypergeometric functions.

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  • $\begingroup$ This is very interesting! XD no doubt something I'll encounter... eventually $\endgroup$
    – iglizworks
    Commented Jul 10, 2021 at 18:29

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