When 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.

Oh 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)?

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!

Thanks! 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_...

Bring radicals

Awesome reference, thank you!

> 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_q...

He 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.

Fascinating, thanks!