I became curious about the following problem from a discussion in another thread: https://www.physicsforums.com/threads/showing-a-polynomial-is-not-solvable-by-radicals.895282/After a bit of study I concluded that the meaning of the assertion below regarding some specific real number rl P has the meaning which follows it. Assertion: "r is not expressible in terms of radicals." Meaning: r is not expressible in terms of a finite application of a collection of operators (+, -, ×, and /, together with any of the n-th roots (where n a positive integer) ), where these operators are applied to integer operands, or to expressions of the same kind.If this meaning is incorrect, I hope someone will correct it. The article https://en.wikipedia.org/wiki/Quintic_functioncontains the following text: An example of a quintic whose roots cannot be expressed in terms of radicals is x5 − x + 1 = 0.This simple example of a Bring-Jerrard quintic equation has one real root, with an approximate value of r = -1.16730397783.This can be seen in the attached PNG file. My QUESTION IS: What would a proof that "r is not expressible in terms of radicals" look like?I have no idea whatever how one would go about proving the non-radical nature of just this one example. Any help would be much appreciated.