[zyklu5]

Let me take this opportunity to post one of the best texts on Galois Theory I have read -- and I had to go through quite a few while preparing for a class.

https://pages.uoregon.edu/koch/Galois.pdf

The subject is developed very naturally and every idea is beautifully motivated. It begins with a quick one chapter intro of Arnold's proof of Abel-Ruffini.

Richard Koch's home page (https://pages.uoregon.edu/koch/) has other examples of his fantastic pedagogy.

[almostgotcaught]

> It begins with a quick one chapter intro of Arnold's proof of Abel-Ruffini.

The key to understanding/motivating Galois theory is Abel-Ruffini, which is a corollary of Galois. And the simplest way to understand that is Arnold's topological proof, which i learned about from this video

https://www.youtube.com/watch?v=RhpVSV6iCko

Watching that video and rolling it around in my head completely demystified Galois theory for me, years after literally 2 semesters of algebra in undergrad. Everything about normal subgroups and commutators and splitting fields and blah blah blah immediately became tangible and obvious. It should be a crime not teach this proof first.

The coverage in Koch's book looks good too - lots of pictures - and funny enough it links to a different youtube video.

Edit: copy-pasting notes I took from the video after watching.

------

The idea is to continuously perturb each of the coefficients of the polynomial along a loop (change each of them from their initial value such that they traverse a path that returns them to that initial value at the end of the path) and study what happens to the roots of the polynomial.

Note, once all coefficients have returned to their original values the entire set of roots also returns to itself, but each root does not necessarily returns to its original value. In general you get a permutation of the set of roots and so in this way we get a mapping between loops of the coefficients and permutations of the roots.

Also note, we can produce coefficient loops that map to any permutation of the roots by permutating the roots and “watching” the coefficients.

Hence, the way to prove Abel-Ruffini is to show that any expression involving the coefficients (ie formula for the roots in terms of the coefficients) returns to itself after the coefficients traverse their loops but the roots do not (and therefore the expression cannot capture all of the roots). For example, an immediate corollary of the construction of the mapping between loops of coefficients and roots is the fact that a general solution involving only -, +, ×, ÷ is not possible; -, +, ×, ÷ are all single-valued and therefore no composition thereof could produce multiple roots.

[zyklu5]

There is indeed a deep connection between what is going on behind Arnold's proof and the classical Galois theory. But it needs quite a bit of sophistication to flesh out properly (not apparent in his famous lectures given to high school kids). There is a Galois theory for Riemann surfaces over algebraic functions where the coverings behave like fields do in the classical correspondence. If any one is interested, check out chapter 3 of Khovanskii's Galois Theory, Coverings and Riemann Surfaces.

[ziofill]

Is the idea of roots swapping places related to Riemann surfaces? A bit like the sqrt function being defined on two copies of the complex plane glued together.

[computator]

> one of the best texts on Galois Theory I have read

What book on group theory or abstract algebra would you recommend to read first to be able to read that text on Galois Theory?

[zyklu5]

If I'm being honest I really don't have a good recommendation for a text in intro abstract algebra. I learned it from Michael Artin's Algebra. Artin is a true master -- along with David Mumford, he was the main apostle for Grothendieck style AG in the US -- but his book was not very easy to learn from.