Elementary properties of the Boolean hull and reduced quotient functors

Abstract

In [12] we proved the following isotropy-reflection principle:Theorem. Let F be a formally real field and let Fp denote its Pythagorean closure. The natural embedding of reduced special groups from Gred(F) into Gred(Fp) = G(FP) induced by the inclusion of fields, reflects isotropy.Here Gred(F) denotes the reduced special group (with underlying group Ḟ/ΣḞ2) associated to the field F, henceforth assumed formally real; cf. [11], Chapter 1, §3, for details.The result proved in [12] is, in fact, more general. For example, the Pythagorean closure Fp can be replaced in the statement above by the intersection of all real closures of F (inside a fixed algebraic closure). Similar statements hold, more generally, for all relative Pythagorean closures of F in the sense of Becker [3], Chapter II, §3.Since the notion of isotropy of a quadratic form can be expressed by a first-order formula in the natural language LSG for special groups (with the coefficients as parameters), this result raises the question whether the embedding ιFFp: Gred(F) ↪ G (Fp) is elementary. Further, since the LSG-formula expressing isotropy is positive-existential, one may also ask whether ιFFp reflects all (closed) formulas ofthat kind with parameters in Gred(F).In this paper we give a negative answer to the first of these questions, for a vast class of formally real (non-Pythagorean) fields F (Prop. 5.1). This follows from rather general preservation results concerning the “Boolean hull” and the “reduced quotient” operations on special groups.