Hilbertianity of fields of power series

Abstract

Let R be a domain contained in a rank-1 valuation ring of its quotient field. Let R[ [X] ] be the ring of formal power series over R, and let F be the quotient field of R[ [X] ]. We prove that F is Hilbertian. This resolves and generalizes an open problem of Jarden, and allows to generalize previous Galois-theoretic results over fields of power series.