Intersections of Real Closed Fields

1. In this paper we wish to study fields which can be written as inter­ sections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hered­ itarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythago­ rean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal(F/F), the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3]. In § 3 a characteriza­ tion is given in terms of finite Galois extensions of the field. This is applied in § 4 to show that these fields suffice to obtain all isomorphism classes of reduced Witt rings (of equivalence classes of anisotropic quadratic forms over a field) with a certain finiteness condition on the rings. In this section we shall briefly outline some of the work other authors have done with these and related classes of fields. Our interest is only in formally real fields, though to study them we shall often have to look at their algebraic extensions. For any formally real field F, we denote by F* the intersection of all the real closed subfields of a fixed algebraic closure F which contain F. These fields have been studied in [6] where they are called "galois order closed" because of the following theorem.