# Decide whether a polynomial has a root

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?

• $A'=\mathbb{R}$ is easily decided using Sturm's theorem and so do all the cases in the last paragraph.
– plop
Oct 15 '20 at 2:23