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
Microsoft Copilot seems to prefer this phrasing -
ReplyDelete“Vieta’s formulas show that the coefficients of a polynomial are symmetric functions of its roots, so any permutation of the roots leaves the coefficients unchanged. But when you write explicit radical expressions for individual roots, nested radicals introduce branch choices that make permuting the roots nontrivial. Only certain permutations—such as complex conjugation when coefficients are real—correspond to simple transformations of the radical expressions. A general quintic formula in radicals would require these radical expressions to accommodate all permutations of the roots, but the symmetry structure of the generic quintic (with Galois group S5) is too large and non‑solvable for this to be possible.”