Undecidability conjecture for Diophantine problems over number fields and rings of algebraic integers

Let $F$ be a number field, meaning a finite algebraic extension of $\mathbb{Q}$, and let $\mathcal{O}$ be a ring of algebraic integers. The Diophantine problem asks whether polynomial equations with coefficients in the given ring or field have solutions in that structure. Undecidability conjecture. The Diophantine problem in $\mathbb{Q}$, as well as in any number field $F$, or any ring of algebraic integers $\mathcal{O}$, is undecidable. This extends the classical undecidability of the Diophantine problem in $\mathbb{Z}$ and is a major conjecture in number theory; its status is not resolved in the supplied text.