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As I mentioned in the other thread, Matiyasevich's theorem implies that it is undecidable whether a system of Diophantine equations over $\mathbb{Z}$ has a solution (Hilbert's 10th Problem). I have to mention some related results here: if $\mathbb{Z}$ is replaced by $\mathbb{F}_p[t]$ then the problem is still not decidable, if replaced by $\mathbb{R}, \mathbb{C},$ or $\mathbb{Q}_p$ then the problem is decidable, and if replaced by $\mathbb{Q}$ or $\mathbb{F}_p((t))$ the answer is not known! (Reference.) I believe it is not even known whether the answer is yes for some number fields but no for others (which would be truly bizarre).