Here's currently the best explanation I have of why there is no general formula in radicals for the roots of a quintic polynomial. It's more for my quick reference than anything else. It may not be totally accurate but maybe it's close.
Vieta's formulas put the roots of a general polynomial equation with unspecified variable coefficients in a position to be swappable, but potentially roots nested inside other roots can pop up, and they can't be easily swapped if they're not conjugates. If a general quintic formula in radicals existed, it would violate this conjugate swapping structure.
I think the biggest pitfall I've gotten caught in is not realizing that the only requirement for SPECIFIC examples where we already know the coefficients is that they respect the nested conjugate rule and don't violate it. Their roots don't have to be all swappable with each other. But when we don't KNOW the coefficients and are dealing with a general variable-coefficient case, we have to allow all roots to be swappable because a specific polynomial may potentially have that property.
These ideas build on a video I uploaded a while back here: https://www.youtube.com/watch?v=qOHkF26EKfg
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