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 anisotropi