DECIDABLE ALGEBRAIC FIELDS

We discuss the connection between decidability of a theory of a large algebraic extensions of ℚ and the recursiveness of the field as a subset of a fixed algebraic closure. In particular, we prove that if an algebraic extension K of ℚ has a decidable existential theory, then within any fixed algebra...

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Bibliographic Details
Published in:The Journal of symbolic logic 2017-06, Vol.82 (2), p.474-488
Main Authors: JARDEN, MOSHE, SHLAPENTOKH, ALEXANDRA
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Existentialism
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Summary:We discuss the connection between decidability of a theory of a large algebraic extensions of ℚ and the recursiveness of the field as a subset of a fixed algebraic closure. In particular, we prove that if an algebraic extension K of ℚ has a decidable existential theory, then within any fixed algebraic closure ℚ̃ of ℚ, the field K must be conjugate over ℚ to a field which is recursive as a subset of the algebraic closure. We also show that for each positive integer e there are infinitely many e-tuples σ ∈ Gal(ℚ)e such that the field ℚ̃(σ) is primitive recursive in ℚ̃ and its elementary theory is primitive recursively decidable. Moreover, ℚ̃(σ) is PAC and Gal(ℚ̃(σ)) is isomorphic to the free profinite group on e generators.
ISSN:0022-4812
1943-5886
DOI:10.1017/jsl.2017.10