1 results
DECIDABLE ALGEBRAIC FIELDS
-
- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 2 / June 2017
- Published online by Cambridge University Press:
- 19 June 2017, pp. 474-488
- Print publication:
- June 2017
-
- Article
- Export citation
-
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.
