I mean both "solvable" and "expressible in radicals" in the same senses as in the Abel-Ruffini theorem. It could be paraphrased "given an irreducible polynomial with one root that is expressible in radicals, are the rest of the roots also expressible in radicals?"
The context for the question is in service of rationalizing the denominators of arbitrary numbers expressed in radicals. Wikipedia says this is done by multiplying the numerator and denominator by all algebraic conjugates of the denominator, which I would think would entail finding the minimal polynomial over the rationals of the denominator, then finding all roots of this polynomial; if the answer to the question is no, this method would sometimes fail. In that case, I would wonder whether the denominator rationalization problem is also not always solvable, or if it might be solvable by another method.