6 ms·
> "even if God told you the answer, you would have no way to write it down" But isn't this also true for generic quadratics/cubics too? Like the solution to x^
by returningfory2 2y ago
> "even if God told you the answer, you would have no way to write it down"
But isn't this also true for generic quadratics/cubics too? Like the solution to x^3-2=0 is cubed_root_of(2), so it seems we can "write it down". But what is the definition of cubed_root_of(2)? Well, it's the positive solution to x^3-2=0...
- impendia 2y agoWhen I say that a fundamental lesson of field theory is that radicals are not really special, this is what I mean. You are thinking in a more sophisticated way than most newcomers to the subject.
- returningfory2 2y agoOh sorry, right. I feel there is an interesting follow-up problem here. The polynomials x^n+a=0 are used to define the "radicals" which is a family of functions F_n such that F_n(a) = real nth root of a = real solution of x^n + a = 0. Using these radicals you can solve all quadratics, cubics and quintics. Now take another collection of unsolvable polynomials; your example was x^5 - x - 1 = 0 and maybe parameterize that in some way such that these polynomials are unsolvable. This gives us another family of functions G_n. What if we allow the G_n's to be used in our solutions? Can we solve all quintics this way (for example)?
- donkeybeer 2y agoBring radicals
- returningfory2 2y agoAwesome reference, thank you!
- impendia 2y agoI'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_...
- itishappy 2y ago> Can we solve all quintics this way (for example)? Nope. Wanna hazard a guess what theory was instrumental in proving that there's no closed form for general quintics? https://en.wikipedia.org/wiki/Galois_theory#A_non-solvable_quintic_example https://en.wikipedia.org/wiki/Galois_theory#A_non-solvable_q...
- donkeybeer 2y agoHe is not wholly wrong, while not solvable in the general case by normal radicals, there is a family of functions, a special "radical" as he said, the Bring radical that solves the quintic generally. Of course as said its not a 5th root, but the solution to a certain family of quintics.
- itishappy 2y agoFascinating, thanks!