Galois groups and complete domains

Consider a domain that is complete with respect to a non-zero prime ideal. This paper proves two Galois-theoretic results about such rings. Using Grothendieck’s Existence Theorem we prove that every finite group occurs as the Galois group of a Galois extension of . This generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if is a Noetherian domain that is complete with respect to a non-zero prime ideal, then every finite group occurs as a Galois group over . This proves the Noetherian case of a conjecture posed by Moshe Jarden.