On Fields of Totally S-adic Numbers

Given a finite set S of places of a number field, we prove that the field of totally S-adic algebraic numbers is not Hilbertian. The field of totally real algebraic numbers Qtr, the field of totally p-adic algebraic numbers Qtot,p, and, more generally, fields of totally S-adic algebraic numbers Qtot,S, where S is a finite set of places of Q, play an important role in number theory and Galois theory, see for example [5, 8, 9, 7]. The objective of this note is to show that none of these fields is Hilbertian (see [3, Chapter 12] for the definition of a Hilbertian field). Although it is immediate that Qtr is not Hilbertian, it is less clear whether the same holds for Qtot,p. For example, every finite group that occurs as a Galois group over Qtr is generated by involutions (in fact, the converse also holds, see [4]) although over a Hilbertian field all finite abelian groups (for example) occur. In contrast, over Qtot,p every finite group occurs, see [2]. In fact, although (except in the case of Qtr) it was not clear whether these fields are actually Hilbertian, certain weak forms of Hilbertianity were proven and used, both explicitly and implicitly, for example in [4, 6]. Also, any proper finite extension of any of these fields is actually Hilbertian, see [3, Theorem 13.9.1]. The non-Hilbertianity of Qtot,p was actually implicitly stated and proven in [1, Examples 5.2] but this result seems to have escaped the notice of the community and was forgotten. We give a short elementary proof (which is closely related to the proof in [1]) of the following more general result. Theorem 1. For any finite set S of real archimedean or ultrametric discrete absolute values on a field K, the maximal extension Ktot,S of K in which every element of S totally splits is not Hilbertian. Note that Ktot,S is the intersection of all Henselizations and real closures of K with respect to elements of S. We would like to stress that S does not necessarily consist of local primes in the sense of [7]. The authors are indebted to Pierre Dèbes for pointing out to them the result in [1]. They would also like to thank Sebastian Petersen for motivation to return to the subject of this note. This research was supported by the Lion Foundation Konstanz and the Alexander von Humboldt Foundation.