Diophantine Undecidability in Some Rings of Algebraic Numbers of Totally Real Infinite Extensions of Q
نویسنده
چکیده
This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of Q, where Hilbert’s Tenth Problem is undecidable.
منابع مشابه
Rings of Algebraic Numbers in Infinite Extensions of Q and Elliptic Curves Retaining Their Rank
We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of Q, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.
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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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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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Let M be a number field. Let W be a set of non-archimedean primes of M . Let OM,W = {x ∈ M | ordpx ≥ 0∀p 6∈ W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particu...
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We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.
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عنوان ژورنال:
- Ann. Pure Appl. Logic
دوره 68 شماره
صفحات -
تاریخ انتشار 1994