I Does anyone really understand what's really going on with this video?

[swampwiz]

If so, I want to have a nice thread where I can ask questions.

 

[Vanadium 50]

If it's not worth five minutes of your time to summarize a video, why is it worth 15 minutes of our time to watch it?

 

[fresh_42]

The proof uses the fact that $A_5$ is simple, i.e. not solvable. Solvability would be necessary for a solution. But solvability can be expressed by the commutator group series. This means fore $A_5$ that the commutator groups are always the entire group, i.e. the degree of the polynomials cannot be lowered.

That's it. The rest is translation work. Have fun!

 

[Stephen Tashi]

If so, I want to have a nice thread where I can ask questions.

The content of the video is worth discussing.

Glancing at https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf , it appears to use the same ideas as the video.

 

[swampwiz]

Evidently there is, via topological methods in the complex plane, a proof of the impossibility of the quintic that does not require one to use abstract algebra & Galois theory.

I think I understand how he was able to have a complex-complex plane mapping of the coefficients & roots of a polynomial so that doing a certain set of path movements for a a pair of roots to the original positions of the other (i.e., a net swap), the corresponding coefficients - all of which depend on all the roots as per the elementary symmetric functions - will move along a return-path. Thus it is possible to have a set of swaps to generate any permutation of the roots while leaving the coefficients the same. Then he talks about "commutators" and "commutators of commutators" and "commutators of commutators of commutators of commutators" ad absudum; these commutators seem to be permutations, and evidently the net permutation from these commutators resolve down from an initial 120 to 60, but then stay at 60, whereas for a size of 4 or less, they resolve down in a way that seems to mirror how Lagrange resolvents are able to get a solution for up to the quartic.

 

[Stephen Tashi]

I think I understand how he was able to have a complex-complex plane mapping of the coefficients & roots of a polynomial so that doing a certain set of path movements for a a pair of roots to the original positions of the other (i.e., a net swap), the corresponding coefficients - all of which depend on all the roots as per the elementary symmetric functions - will move along a return-path.

Referring to https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf

The first assertion:

It is possible to have a path that traverses a closed loop in the space of coefficients and implies a path in the space of roots that is not closed. Specifically: taking $t$ to be the variable that parameterizes the path, there exists a path $(a(t),a(t),c(t))$ in the space of coefficents such that $( a(0), b(0), c(0)) = (a(1),b(1),c(1))$ and whose corresponding path $(x(t),y(t))$ in space of roots has $(x(0),y(0)) = (r_1,i_i)$ and $(x(1),y(1)) = (r_2, i_2)$ with $r_1 + i_1 i$ and $r_2 + i_2 i$ being two distinct roots of the equation $a(1)x^2 + b(1)x + c(1) = 0$

Hence:

Proposition 1. There does not exist any continuous function from the space of quadratic polynomials to $\mathbb{C}$ which associates to any quadratic polynomial a root of that polynomial

i.e. There is no continuous function $f(A,B,C)$ of the coefficients of a (general) quadratic polynomial $Ax^2 + Bx + C$ such that $f(A,B,C)$ is a root of the polynomial.

i.e. You cannot solve for a root of a (general) quadratic equation by using a formula that specifies a continuous function.

Do you understand why the proposition follows from the behavior of the paths?

 

[swampwiz]

Referring to https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf

The first assertion:

It is possible to have a path that traverses a closed loop in the space of coefficients and implies a path in the space of roots that is not closed. Specifically: taking $t$ to be the variable that parameterizes the path, there exists a path $(a(t),a(t),c(t))$ in the space of coefficents such that $( a(0), b(0), c(0)) = (a(1),b(1),c(1))$ and whose corresponding path $(x(t),y(t))$ in space of roots has $(x(0),y(0)) = (r_1,i_i)$ and $(x(1),y(1)) = (r_2, i_2)$ with $r_1 + i_1 i$ and $r_2 + i_2 i$ being two distinct roots of the equation $a(1)x^2 + b(1)x + c(1) = 0$

Hence:
i.e. There is no continuous function $f(A,B,C)$ of the coefficients of a (general) quadratic polynomial $Ax^2 + Bx + C$ such that $f(A,B,C)$ is a root of the polynomial.

i.e. You cannot solve for a root of a (general) quadratic equation by using a formula that specifies a continuous function.

Do you understand why the proposition follows from the behavior of the paths?

Yes, I have been working on grokking this, and I think I see the point. Basically if you start with the coefficients and one of the roots and then continuously modulate one of the coefficients (the best one seems to be the one whose order is 1 less than the degree) and have it do a cycle (i.e., a revolution about the origin) and return to the starting point, that selected root is at the negative of where it was originally, and thus there would be 2 values for the same input value, which would be impossible for a continuous function - and since the question is whether it is a continuous function, only the root that is on its own continuous path would suffice, so the fact that the other one there doesn't matter.

 

[Stephen Tashi]

Yes, I have been working on grokking this, and I think I see the point. Basically if you start with the coefficients and one of the roots and then continuously modulate one of the coefficients (the best one seems to be the one whose order is 1 less than the degree) and have it do a cycle (i.e., a revolution about the origin) and return to the starting point, that selected root is at the negative of where it was originally, and thus there would be 2 values for the same input value, which would be impossible for a continuous function - and since the question is whether it is a continuous function, only the root that is on its own continuous path would suffice, so the fact that the other one there doesn't matter.

The particular feature that a closed path in the coefficient space can correspond to a path in the root space that maps a root to its negative is interesting. The more general idea is that a closed path in the coefficient space can correspond to a path in the root space that begins at one root and ends at a different root (whether the different root is the negative of the initial one or not).

If the mapping from the coefficient space to the root space was continuous then a closed path in the coefficient space should map to a closed path in the root space - by analogy to the way that the limit of a continuous function of several variables at a point should be the same as any limit taken along a path in the domain leading to that point.

That's the way I understand it - which is admittedly not in a precise and rigorous way.

(In further web searching on the phrase "Arnold's proof of Abel's theorem", I found references to a book that covers the ideas in the video: V.B. Alekseev, Arnold's Proof in Problems and Solutions. The book is based on lectures to "Russian high school students"!
I haven't read it. )

I think the next task is to understand (at least intuitively) mappings from the coefficient space to the root space that allow the use of a complex valued $\sqrt[n]{()}$ function and thus have discontinuities of the particular kind caused by that type of function.

A algebraic formula employing $\sqrt[n]{()}$ can involve using several $\sqrt[n]{()}$ functions at intermediate stages of computing the final result of the formula. So analyzing how such a formula maps closed paths in the coefficient space to paths in the root space is not as simple as analyzing how a single use of $\sqrt[n]{()}$ affects closed paths.

A algebraic formula can be visualized as a sequence of mappings. We can think of closed path in the coefficient space being mapped to a different path in each step in this sequence. (The dimensions of the domains and ranges of maps in different steps aren't clear to me. Can we visualize each map except the final one as a map from the coefficient space into itself? )

A closed path in the coefficient space is associated with a unique permutation of the roots. The general idea must be that compositions of maps involving $\sqrt[n]{()}$ can only accomplish certain permutations of the roots, but there exist paths in the coefficient spaces of some 5th degree polynomials that accomplish different permuations.