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Comment by enriquto | Hacker News Reader

Well, some quintics are solvable, aren't they? (like x^5-1=0). Maybe for each particular quintic there is a different formula that solves it. You just prove that there's no formula that works in all cases when you plug the coefficients at the same place.

It would be more satisfying (and more general) to exhibit a specific unsolvable quintic.

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Yes, that's correct. That's the big disadvantage to the topological method I talk about (developed by Vladimir Arnold); it can only prove stuff about the general nth-degree formulas, whereas Galois theory can prove stuff about specific polynomials (e.g., that x^5 – x – 1 is unsolvable in radicals).

On the other hand, the topological method can be easily extended to prove stuff about formulas including other continuous, single-valued operations, like exponential and trigonometric functions, and Galois theory has a harder time with this.

> That's the big disadvantage to the topological method I talk about (developed by Vladimir Arnold); it can only prove stuff about the general nth-degree formulas, whereas Galois theory can prove stuff about specific polynomials (e.g., that x^5 – x – 1 is unsolvable in radicals).

Can these topological methods be generalized to arbitrary fields?

I'm a bit out of my depth here, but I think you can generalize it to differential fields, where you want to figure out e.g. which functions have a closed-form integral. A reference (the only reference?) would be Topological Galois theory: https://www.springer.com/us/book/9783642388705

All quintics have solutions, but there is no general quintic formula.

Of course, but in this game "solution" means "a number that can be expressed as a composition of radicals and elementary operations".

Not true, there's only no general quintic formula that employs a specific, restricted set of operations.

In practice, there is a general quintic formula. It just needs one extra operation.

What other operation? If we don't know what it is, how do you know it needs just one?

Does that matter?

If you are free to add any additional operation, this whole thing becomes meaningless. You can simply define your operation "NewOp(a0,a1,..,a4)" to something like "the smallest root of the quintic a0+a1x+...+a4x^4+x^5".

(Here, "smallest" can be anything as long as it is a completely defined tie-breaker, such as: the value with the smallest real part, and among those the one with the smallest imaginary part.)

I think it's something like p(x) = smallest positive solution of z^5 + z - x. Compare sqrt(x) = smallest positive solution of z^2 - x.

But it's a _long_ time since I've done Galois theory and I can't find a decent math exchange answer for it right now, so don't treat this as gospel.

Yeah, I think you're thinking of a Bring radical: https://en.wikipedia.org/wiki/Bring_radical

Edit: Oh, but that only lets you solve some quintics. https://news.ycombinator.com/item?id=14686886 describes the functions you need to solve all quintics.

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