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DECIDABLE ALGEBRAIC FIELDS

Published online by Cambridge University Press:  19 June 2017

MOSHE JARDEN  and
ALEXANDRA SHLAPENTOKH
MOSHE JARDEN
Affiliation:
SCHOOL OF MATHEMATICS TEL AVIV UNIVERSITY RAMAT AVIV TEL AVIV 6139001, ISRAEL E-mail: jarden@post.tau.ac.il
ALEXANDRA SHLAPENTOKH
Affiliation:
DEPARTMENT OF MATHEMATICS EAST CAROLINA UNIVERSITY UNIVERSITY GREENVILLE NC 27858-4353, USA E-mail: shlapentokha@ecu.edu
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Abstract

We discuss the connection between decidability of a theory of a large algebraic extensions of ${\Bbb Q}$ 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 ${\Bbb Q}$ has a decidable existential theory, then within any fixed algebraic closure $\widetilde{\Bbb Q}$ of ${\Bbb Q}$ , the field K must be conjugate over ${\Bbb Q}$ 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 $\sigma\in {\text{Gal}}\left( {\Bbb Q} \right)^e $ such that the field $\widetilde{\Bbb Q}\left( \sigma\right)$ is primitive recursive in $\widetilde{\Bbb Q}$ and its elementary theory is primitive recursively decidable. Moreover, $\widetilde{\Bbb Q}\left( \sigma\right)$ is PAC and ${\text{Gal}}\left( {\widetilde{\Bbb Q}\left( \sigma\right)} \right)$ is isomorphic to the free profinite group on e generators.

Keywords

12E3003C0703D35decidable theoryrecursive subsets

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 474 - 488
Copyright
Copyright © The Association for Symbolic Logic 2017 

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