Short proof of Abel's theorem that 5th degree polynomial equations cannot be solved
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as an accessible, visual walkthrough of Arnold's proof of the Abel-Ruffini theorem, both as a reference and as the source of follow-up questions.
Recommended by 2 people across 2 different months, 2015 to 2024.
What it is
A roughly 16-minute lecture by Boaz Katz presenting a shortened version of Vladimir Arnold's topological proof that the general quintic cannot be solved in radicals. It tracks how roots move in the complex plane as coefficients are varied along loops, and assumes familiarity with complex numbers.
Cited in 1 Wikipedia article, including Abel–Ruffini theorem.
What people said
Recommends M.B. · 2021-02 · Mathematics Stack Exchange · on “There is no solution in radicals to general polynomial equations of degree five.”
@JackM: Arnold's proof of the insolvability of the quintic requires no more than basic complex analysis (his proof was in fact aimed at high school students). See e.g. this video (and other similar videos on yt): [this video]
Recommends Chris Daniel · 2024-07 · Mathematics Stack Exchange · on “Help needed with understanding Arnold's proof of Abel-Ruffini Theorem”
I am trying to understand Arnold's proof with this document [LINK] and this video [this video] . In section 5 the existence of the permutation (1 2 3 4 5) in the set of commutator loops is used to conclude that any quintic formula must contain nested radicals. This reasoning is then repeated again and again to obtain Theorem 9 . I think I have understood the reasoning but just want to check here. The same logic is used in section 4 to show any cubic formula must contain nested radicals. My understanding so far is this: Am I correct in saying that the proof of exercise 1 is that the point $p$ ends up without any phase change after travelling a commutator of loops and is therefore on the same branc
Cites Paramanand Singh · 2015-07 · Mathematics Stack Exchange · on “Unsolvability of a Quintic and its link with "Simplicity" of $A_{5}$”
I know this criterion of solvability using derivative (commutator subgroup). This is the essentially the same idea behind Arnold's proof given in [this video] What I want to understand is not so much the proof for unsolvability of $A_{5}$, but rather the link of theorem in my post with the unsolvability / simplicity of $A_{5}$.
Cites Noah · 2016-07 · Mathematics Stack Exchange · on “Arnold's proof of Abel's theorem”
I'm seeking help understanding this video [this video] . The author considers the equation $ax^5+bx^4+cx^3+dx^2+ex+f = 0$ and shows both the coefficients $a, b$... and solutions $x_1, x_2$... in the complex plane. The author claims that if the coefficients are varied, moving them along short loops so they return to their original values and the solutions don't return to their original values, but instead exchange places, an expression for the solution cannot be found. What is the significance of the solutions not returning to themselves?