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Topological proof for the unsolvability of the quintic

Mathematics Asked on November 26, 2021

Sorry, but in order to ask the question, you will have to view this video
http://drorbn.net/dbnvp/AKT-140314.php.
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 this this video. If you have a reference here, please let me know!

Here are some more references I have found:

One Answer

When you have a submanifold of dimension $k$ inside an (ambient) manifold of dimension $n$, we define its codimension as $n-k$. Now you know that the discriminant as a function on $mathbb{C}^6$ is given by some function (choose your favourite formula, mine is the Vandermonde determinant), but any formula you use should be smooth, and it gives you a function $mathbb{C}^6 to mathbb{C}$. The "bad" points are where the discriminant is $0$ and you can check that $0$ is a regular value, so $f^{-1}(0)$ is something $5$-(complex) dimensional i.e. something codimension $1$. As for why it didn't matter, I can't really say anything besides I didn't see any issue in throwing out these points in the ensuing discussion (which is probably what Bar-Natan meant you'll see it doesn't matter).

Answered by Osama Ghani on November 26, 2021

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