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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?

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

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