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Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, themaximal abelian extension of $\mathbb{Q}$. We show that there exists analgorithm to decide whether a system of equations and inequations, with integercoefficients, has a solution in $\mathbb{Z}^{ab}$ modulo every rational prime.

Diophantine problems over $\mathbb{Z}^{ab}$ modulo prime numbers