Why There's 'No' Quintic Formula (proof without Galois theory)

What people said

Strong rec Reinhard Meier · 2024-07 · Mathematics Stack Exchange · on “Question on Arnold's proof of the Abel-Ruffini Theorem”

I really recommend the YouTube Video Why there's 'No' Quintic Formula [this video] by "not all wrong". It specifically explains Vladimir Arnold's argument for the non-existence of a general algebraic formula for solving quintic equations.

Recommends Hans Lundmark · 2021-10 · Mathematics Stack Exchange · on “Arnold's Topological Proof for the Insolvability of the Quintic: image of $f^\alpha(p)$ as $p$ traverses a commutator loop”

There's a really excellent video about Arnold's proof here [this video] (by math.SE user not all wrong [LINK] , by the way). Maybe that can help you sort it out?

Recommends levitopher · 2022-12 · Mathematics Stack Exchange · on “quintic solution to noncommutative polynomial”

So, we know there are generally no solutions to an arbitrary quintic polynomial (EDIT from comments: Of course, I mean you can't write the roots using radicals, not that the roots don't exist). I don't know Galois theory, so I understand this result by thinking about permutations of roots and coefficients ( this video is awesome [this video] !). To me it seems to boil down to the fact that when you write a polynomial in terms of coefficients and roots: $$x^5 + ax^4 + bx^3+cx^2+dx+e=(x-x_1)(x-x_2)(x-x_3)(x-x_4)(x-x_5)$$ There is a natural ordering of coefficients, but not a natural ordering of the roots (you can permute the $x_i$ 's and get the same set of coefficients, but if you switch the coeff

Cites D.R. · 2022-05 · Mathematics Stack Exchange · on “Why does Galois theory most naturally take place in the context of fields?”

At least as far as I can tell, historically Galois theory [LINK] was a more computational tool than it appears now, and [LINK] , How Did Galois Understand the Galois Group? [LINK] and Intuition behind looking at permutations of the roots in Galois theory [LINK] seem to say that the "fundamental idea of Galois theory" is to study permutations of the roots, which was first done via symmetric polynomials. The often presented motivation of Galois theory, the insolubility of the quintic, can also be presented without "modern Galois theory", as in this wonderful video on Arnold's proof: [this video] . This video also uses the "fundamental idea of Galois theory", i.e. studying permutations of the roots,