BSS-model Computational complexity of finding the roots of a polyomial

Question

I'm currently dealing with a problem for which I could show that an exact algorithm would imply a general algorithm for finding the real (but not complex) roots of an arbitrary univariate polynomial with real coefficients.

Now, what exactly are the implications for my problem?

It is well-known that there is no formula that describes the roots of a polynomial of degree $\geq 5$ and, as far as I know, there is no general algorithm that computes the roots exactly. But are there results stating that the problem is computationally intractable in general? Or is it a common assumption? My question is not aiming at existence results or approximation algorithms but at the computational complexity of the problem and possible implications for my problem.

Note: My algorithm works in the Blum-Shub-Smale model, so the possibly infinite representations of the solutions don't bother me.