Suppose you have some polynomial $p(x)$ with rational coefficients in which at least one root is unsolvable by radicals, does this imply that all other roots of $p(x)$ are unsolvable by radicals?
I have been thinking about algebraic numbers for a while, and I found this question interesting. I tried a few examples, and I tried to think to a solution but all I concluded was this below.
If the constant factor is zero, and the polynomial has an unsolvable root, then I have an answer to the question since if the constant is zero, then it must have a root at zero.
This also implies that if the answer to my question is yes, then any polynomial with zero as the constant term, is solvable by radicals.