Abstract
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 the fields to show the first-order undecidabilitv. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. 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 qm. (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 \(\{ {\zeta _{{p^l}}}|l \in {Z_{ > 0,}}P \ne q\) is any prime such that qm +1 ∧(p — 1)}. (3) The first-order theory of Any Abelina Extension of Q With Finitely Many Rational Primes is undecidable and rational integers are definable in these extensions.
We also show that under a condition on the splitting of one rational Q generated elliptic curve over the field in question is enough to have a definition of Z and to show that the field is undecidable.
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The research for this paper has been partially supported by NSF grants DMS-0650927 and DMS-1161456, and a grant from John Templeton Foundation. The author would also like to thank Carlos Videla for helpful comments.
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Shlapentokh, A. First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers. Isr. J. Math. 226, 579–633 (2018). https://doi.org/10.1007/s11856-018-1708-y
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DOI: https://doi.org/10.1007/s11856-018-1708-y