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

Alexandra Shlapentokh
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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.

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BibTeXRIS

Alexandra Shlapentokh. 2006-06-15. Diophantine Definability and Decidability in the Extensions of Degree 2 of Totally Real Fields. https://arxiv.org/abs/math/0606368

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