Lemma for the Fundamental Theorem of Galois Theory

> 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.

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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.

Me too! But I can't really remember why. The proof was impressive.

Galois theory is so useful that, besides its fundamental importance in algebra, it birthed the whole subject of Galois connections, which crop up all over the place, including in theoretical CS: https://en.wikipedia.org/wiki/Galois_connection .

Similarly, Galois theory was a penny dropping moment in my understanding of mathematics and has remained surprisingly useful late into by career, as software has become increasingly "mathy".

The html source is beautiful, did you write it by hand?

Thank you for sharing your notes—everyone needs a different push/viewpoint to make key concepts click, and, the more expositions are out there, the more likely someone will be to find the one that's right for them!

Is M^* Stewart's notation? I'm much more used to seeing it used for a dual space than a group of automorphisms. Conventions aside, it's a bit unfortunate in that it doesn't specify the field on which we're acting! I think that the notation Aut(L/M), or Gal(L/M) for a Galois extension, is more common.

I’m inclined to agree, but who knows… maybe he just happened to have his big insight early on, and if he’d lived longer he’d never have done anything quite as significant. Plenty of people do great things only once.

The thing with mathematics is there are often many different ways to prove things leaving no universally agreed upon graph of true mathematical statements in terms of which helps proving which! That being said, you might want to check out ProofWiki [1], an online collaborative project featuring mathematical proofs consistently linked together.

[1] https://proofwiki.org/wiki/Main_Page

You mean like a dependency tree? There isn’t a comprehensive one — it would be way too big and we don’t have all mathematical knowledge written down in one place anyway.

But if you’re interested in any given classical result, if you ask I’m sure a mathematician can give you a rough idea of what the path back down to first principles is. If you’re not literally interested in starting by assuming the existence of the empty set, say, then opening any introductory book on the topic will give you an idea. Just look at the proof of the result and follow its references back. It won’t go that deep.