  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.
