Bauer's Theorem (a simple consequence of the Chebotarev Density Theorem) states that a finite Galois extension K of an algebraic number field F is uniquely determined (as a subield of some fixed algebraic closure of F) by the set of primes of F which split completely in K. Thus knowing all possible Galois groups is the same as knowing all possible splitting laws in finite Galois extensions. Being able to describe these splitting laws in some explicit fashion is basically "nonabelian reciprocity", which is THE most important problem in algebraic number theory, so the "inverse Galois problem" is of FUNDAMENTAL importance to all number theorists.