Definability of the ring of integers in some infinite algebraic extensions of the rationals

Let K be an infinite Galois extension of the rationals such that every finite subextension has odd degree over the rationals and its prime ideals dividing 2 are unramified. We show that its ring of integers is first‐order definable in K . As an application we prove that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb Q}(\lbrace \cos (2\pi /\ell ^n) : \ell \in \Delta , n \in {\mathbb N}\rbrace )$\end{document} together with all its Galois subextensions are undecidable, where Δ is the set of all the prime integers which are congruent to −1 modulo 4.