Short proof that 5th degree polynomial equations cannot be solved
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users point people to it as the simplest accessible route to the Abel-Ruffini theorem, and often cite it when asking follow-up questions about Arnold's topological argument.
Recommended by 2 people across 2 different months, 2013 to 2020.
What it is
A 26-minute lecture by Boaz Katz giving a shortened, modified version of Vladimir Arnold's topological proof that the general quintic cannot be solved by radicals. It uses only basic complex numbers and loops of roots, avoiding Galois theory.
What people said
Recommends Boaz Katz · 2013-12 · Mathematics Stack Exchange · on “Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?”
The topological proof by Arnold is simpler to understand. A shortened version is explained in [this video] This requires (only) knowledge of complex numbers and their roots.
Recommends Philipp · 2020-07 · Mathematics Stack Exchange · on “Topological proof for the unsolvability of the quintic”
Sorry, but in order to ask the question, you will have to view this video [LINK] . Here a topological proof for the unsolvability of the quintic is given, based on ideas of Vladimir Arnold. It's amazing because it does not require Galois Theory! And the proof takes only about 15 minutes in the video. (min 6 - min 22) Here is my question: At min 7:10 Dror Bar-Natan removes all coefficients which lead to multiple roots, and says something like "It's a codimension 1 thing that we remove". This is not quite clear to me. What does that mean? Unfortunately I have not found an exposition of this proof, which is written down in a paper or a book, and which is as clear and easy as the exposition is t
Cites user2167741 · 2019-09 · Mathematics Stack Exchange · on “Where exactly Riemann surfaces and monodromy groups appear in this video?”
This is regarding the proof of the unsolvability of the quintic equation (Abel-Ruffini theorem) which is explained really well in this short video [this video] . I also understand that the original proof [LINK] by Arnold uses Riemann surfaces and Monodromy groups. I am wondering where exactly the Riemann surfaces pop in this video? Also, it will be great if someone could intuitively explain what monodromy groups are in the context of this video. Thanks!
Cites user5768239 · 2020-05 · Mathematics Stack Exchange · on “Necessity of nested radicals for cubic equation root formula - Arnold's proof of Abel's theorem”
I watched Boaz Katz' youtube video [this video] and read Leo Goldmakher's paper [LINK] . They both are saying that a commutator does not change a value in a formula with one root of a continuous function $\sqrt{f_1}$ . And if we want to get a different value, we need nested radicals. That's why formula Cardano have to have nested radicals. My question is why would commutator change value of nested radicals? Any examples? My understanding is when the commutator is applied to one radical of a continuous function, arg change will go positive and negative eventually adding up to zero because the two closed paths in the commutator are taking in both directions. But if total arg change of one radical o