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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.