Definability and decidability in infinite algebraic extensions
نویسندگان
چکیده
منابع مشابه
First Order Decidability and Definability of Integers in Infinite Algebraic Extensions of Rational Numbers
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 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 ...
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Let M be a number field, and WM a set of its non-Archimedean primes. Then let OM,WM = {x ∈M | ordt x ≥ 0, ∀t 6∈ WM}. Let {p1, . . . , pr} be a finite set of prime numbers. Let Finf be the field generated by all the pji -th roots of unity as j → ∞ and i = 1, . . . , r. Let Kinf be the largest totally real subfield of Finf . Then for any ε > 0, there exist a number field M ⊂ Kinf , and a set WM o...
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We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of 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...
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متن کاملRelative decidability and definability in henselian valued fields
Let (K, v) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f(x) ∈ K[x], what is v(f(x))? Two applications are given: fir...
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ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 2014
ISSN: 0168-0072
DOI: 10.1016/j.apal.2014.02.001