Let $A$ be a ring such that all elements of $A$ are complex computable numbers. I'm interested in knowing whether the decision problem that asks, given $P\in A[X]$, if $P$ has a root in $A$ is decidable.
I know that this problem is decidable for the following rings:
- $\mathbb{Z}$, since we can easily bound the absolute value of any root of $P\in \mathbb{Z}[X]$ ;
- the field of fractions of a ring $A$ for which the problem is decidable, by Gauss lemma ;
- any algebraically closed field, in particular the field of computable complex numbers ;
- the field of computable real numbers, as a corollary of Sturm's theorem, as pointed out in this comment ;
Is there some general result on the decidability of this class of problems, maybe by making extra assumptions on $A$? Do we know if there are rings for which this decision problem is undecidable?