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

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