Subfields of a real closed field of countable codimension

Abstract

Let R be a real closed field and let K be a subfield of R such that R/K is a proper algebraic extension. The main result of this paper (Theorem 2.6) states that there exists {Kn:n∈N} a countable family of countable codimension subfields of R containing K such that Ks⊆Kt if s∣t and R=⋃n∈NKn. Among other consequences of this result, it is shown that (Corollary 3.1) every real closed field contains a countable family of countable codimension subfields and (Proposition 3.7) if F is the family of all countable codimension subfields of a real closed field, then ⋂E∈FE=Q.