Consider a polynomial defined by its roots:
\begin{equation} P(z; \mathbf{S}) = \Pi_{\theta_j \in \mathbf{S}} (z - \exp({2 \pi i \theta_j}) ) \end{equation}
where $\mathbf{S}$ is a set of numbers. The roots to this polynomial are locations on the unit circle. Let's consider two sets: the set of all rationals $\mathbf{Q}$ and the set of all real numbers $\mathbf{R}$, each between zero and one. While $\mathbf{Q}$ is infinite but countable, $\mathbf{R}$ is not countable, so I'm not sure that this is even a valid polynomial. Nonetheless I was wondering about the following question:
Assume you had a computer program $C(n)$ that could converge to the nth root of this polynomial, but you're not sure if it is $P(\mathbf{Q})$ or $P(\mathbf{R})$. My guess is that you could perform a countably infinite number of computations to rule out or confirm $P(\mathbf{Q})$. Is there any finite computation that you could do to determine what the underlying set is?