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Definability of the ring of integers in some infinite algebraic extensions of the rationals
Author(s) -
Fukuzaki Kenji
Publication year - 2012
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201110020
Subject(s) - rational number , mathematics , undecidable problem , ring (chemistry) , combinatorics , ring of integers , order (exchange) , prime (order theory) , algebraic number , galois group , prime number , modulo , integer (computer science) , discrete mathematics , algebraic number field , mathematical analysis , chemistry , organic chemistry , decidability , finance , computer science , programming language , economics
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.