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Preprint Number 603

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603. Alexandra Shlapentokh
Divisibility of Order and First Order Definability and Decidability in Infinite Algebraic Extensions of Rational Numbers

Submission date: 2 July 2013.


We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of these fields to show first-order undecidability. In particular, we show that the following propositions hold.
(1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by q^m.
(2) Given a prime q, and an integer m>0, algebraic integers are definable in a cyclotomic extension (and any of its subfields) generated by any set \xi_{p^{\ell}}| \ell \in Z_{>0}, p \not=q is any prime such that q^{m+1}\not | (p-1)}.
(3) The first-order theory of any abelian extension of Q with finitely many ramified rational primes is undecidable.

Mathematics Subject Classification: 11U05

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Full text arXiv 1307.0743: pdf, ps.

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