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

We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of 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 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 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.