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By lupire 2025-12-1416:47 Boat Katz's OG video from 11 years ago, possibly the oldest video of this proof on YouTube. https://m.youtube.com/watch?v=RhpVSV6iCko
Vladimir Arnold famously taught a proof of the insolubility of the Quintic to Moscow Highschool students in the 1960s using a concrete, low-prerequisite approach. His lectures were turned into a book Abel’s Theorem in Problems and Solutions by V.B. Alekseev which is available online here: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/abel.pdf. He doesn't consider Galois theory in full generality, but instead gives a more concrete topological/geometric treatment. For anyone who wants to get a good grip on the insolubility of the quintic, but feels overwhelmed by the abstraction of modern algebra, I think this would be a good place to start.
By sesm 2025-12-010:03 This book is also an accessible introduction to group theory, I managed to work through the first half of the book when I was 15 y.o.
By CamperBob2 2025-12-011:394 reply Looks like a nice book, but what's up with his assertion on page 148 (164 of the .pdf) that the integers don't form a group under addition?
If he defines integers as "natural numbers excluding zero," that seems goofy and nonstandard but also interesting. Is that a Russian-specific convention?
By steppi 2025-12-012:22 It seems like a typo where "integers" is used when the intention was to write "natural numbers". That is the solution to exercise 194 part a) which asked if the set of natural numbers is a field.
By CiaranMcNulty 2025-12-0112:42 Whether 0 is a natural number is still fairly ambiguous; I remember being taught (1990s UK) to be specific about which definition was being used, or to prefer another name such as 'positive integers' or 'non-negative integers'
By adgjlsfhk1 2025-12-012:42 notation has unified dramatically since the 1960s.
By Iwan-Zotow 2025-12-013:27 Yes, I think there were copies circulated (not xerox but blueprints)
I learnt this subject from the book Galois Theory, 5th ed. by Ian Stewart. Quoting from page 177:
Theorem 15.10. The polynomial t⁵ - 6t + 3 over ℚ is not soluble by radicals.
As you can see, this theorem occurs in Chapter 15. So it takes fourteen chapters before we reach here. It takes a fair amount of groundwork to reach the point where the insolubility of a specific quintic feels natural rather than mysterious.
To achieve this result, the book takes us through a fascinating journey involving field extensions, field homomorphisms, impossibility proofs for ruler and compass constructions, the Galois correspondence, etc. For me, the impossibility proofs were the most interesting sections of the book. Before reading the book, I had no idea how one could even formalise questions about what is achievable with a ruler and compass, let alone prove impossibility. Chapter 7 explains this beautifully and the algebraic framework that makes those proofs possible is very elegant.
By the time we reach the section about the insoluble quintic, two key results have been established:
Corollary 14.8. The symmetric group S_n is not soluble for n ≥ 5.
Theorem 15.8. Let f be a polynomial over a subfield K of ℂ. If f is soluble by radicals, then the Galois group of f over K is soluble.
The final step is then quite neat. We show that the Galois group of f = t⁵ - 6t + 3 over ℚ is S₅. Corollary 14.8 tells us S₅ is not soluble. By the contrapositive of Theorem 15.8, f is not soluble by radicals.
Obviously whatever I've written here compresses a huge volume of work into a short comment, so it cannot capture how fascinating this subject is and how all the pieces fit together. But I'll say that the book is absolutely wonderful and I would highly recommend it to anyone interested in the subject. The table of contents is available here if you want to take a look: https://books.google.co.uk/books?id=OjZ9EAAAQBAJ&pg=PT4
Two small warnings: The book contains a fair number of errors which can be confusing at times, though there are plenty of errata and clarifications available online. And unless you already have sufficient background in field homomorphisms and field extensions, it can take several months of your life before you reach the proof of the insoluble quintic.
By lupire 2025-12-1416:34 The OP proof is the Abel-Ruffini theorem which requires far less abstract algebra.
