Hilbert's Tenth Problem and the Inverse Galois Problem.

We show that the inverse Galois problem over global fields can be reduced to Hilbert's tenth problem over $\mathbb{Q}$ (and consequently over $\mathbb{Z}$); that is, if there is an algorithm to decide the existence of a solution for an arbitrary polynomial equation over $\mathbb{Q}$, then there is an algorithm to decide whether a certain group $G$ appears as the Galois group of a finite extension of a given global field $K$.