Is there a complexity viewpoint of Galois' theorem?

Question

• Galois's theorem effectively says that one cannot express the roots of a polynomial of degree >= 5 using rational functions of coefficients and radicals - can't this be read to be saying that given a polynomial there is no deterministic algorithm to find the roots?

• Now consider a decision question of the form, "Given a real rooted polynomial $p$ and a number k is the third and the fourth highest root of $p$ at least at a gap of k?"

A proof certificate for this decision question will just be the set of roots of this polynomial and that is short certificate and hence it looks like $NP$ BUT isn't Galois theorem saying that there does not exist any deterministic algorithm to find a certificate for this decision question? (and this property if true rules out any algorithm to decide the answer to this question)

So in what complexity class does this decision question lie in?

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All NP-complete questions I have seen always have a trivial exponential time algorithm available to solve them. I don't know if this is expected to be a property which should always be true for all NP-complete questions. For this decision question this doesn't seem to be true.