Diophantine Definability and Decidability in the Extensions of Degree 2 of Totally Real Fields

Abstract

We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for {\it all} extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of ``small'' and ``big'' subrings of number fields in the infinite algebraic extensions of $\mathbb Q$, previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes.