3 ms·
I'm not sure I understand your question exactly, but I am fairly certain that not all quintic fields can be solved by the combination of (1) radicals, i.e. taki
by impendia 2y ago
I'm not sure I understand your question exactly, but I am fairly certain that not all quintic fields can be solved by the combination of (1) radicals, i.e. taking roots of x^n - 1, and (2) taking roots of x^5 - x - 1. I don't have a proof in mind at the moment, but I speculate it's not too terribly difficult to prove.
If I'm correct, then the proof would almost certainly use Galois theory!
- returningfory2 2y agoThanks! I should have made it more explicit that we would need some family of quantic equations, not just x^5 - x - 1. And looks like from another reply the answer is yes? https://en.wikipedia.org/wiki/Bring_radical#Solution_of_the_general_quintic https://en.wikipedia.org/wiki/Bring_radical#Solution_of_the_...