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I am well aware of the fact that a general quintic polynomial cannot be solved in radicals. However, is there a known way to obtain a formula with other functions (e.g. infinitely nested radicals)?

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    $\begingroup$ You might take a look at arxiv.org/pdf/1308.0955.pdf $\endgroup$ Oct 27, 2021 at 22:03
  • $\begingroup$ Cf. this Math Overflow question $\endgroup$ Oct 27, 2021 at 22:10
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    $\begingroup$ What would be the meaning of infinitely nested radicals? Sounds like the limit of some sequence? Are you working over the reals? You need something extra from the field for the concept of a limit to make sense. $\endgroup$ Oct 28, 2021 at 3:11

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Solving the general quintic using infinitely nested radicals? Of course.

$$x^n=a+x$$

$$x = \sqrt[n]{a+x}$$

By an iterative process,

$$x =\sqrt[n]{a+\sqrt[n]{a+\sqrt[n]{a+\sqrt[n]{a+x\dots}}}}$$

Since the general quintic can be reduced to the one-parameter Bring-Jerrard form, then just substitute $n=5$ to the formula.

P.S. Surprisingly, the two-parameter form $x^n+x^2+ax+b = 0$ can be solved by infinitely nested radicals as well. See this post.

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  • $\begingroup$ If you used $n$th roots of unity, while taking the $n$th root on both sides, would that get you all the $n$ roots of the polynomial? $\endgroup$ Aug 11, 2023 at 12:02
  • $\begingroup$ @TymaGaidash: Ideally, it should by affixing the correct $n$th root of unity, but I haven't tried it yet. $\endgroup$ Aug 11, 2023 at 12:28

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