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All quintics have solutions, but there is no general quintic formula.
by dullcrisp 9y ago
All quintics have solutions, but there is no general quintic formula.
- enriquto 9y agoOf course, but in this game "solution" means "a number that can be expressed as a composition of radicals and elementary operations".
- moomin 9y agoNot 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.
- popcorncolonel 9y agoWhat other operation? If we don't know what it is, how do you know it needs just one?
- vog 9y agoDoes 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.)
- moomin 9y agoI 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.
- akalin 9y agoYeah, I think you're thinking of a Bring radical: https://en.wikipedia.org/wiki/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 https://news.ycombinator.com/item?id=14686886 describes the functions you need to solve all quintics.