2015/02/13 by Moshe Jarden, Jarden, Moshe, Alexandra Shlapentokh +1
Mathematics · #12E30 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Number Theory (math.NT) #math.LO #math.NT #msc:12E30
paper · pdf · doi:10.48550/arxiv.1502.03885
arxiv created 2015/02/13 · openalex publication_date 2015/02/13 · arxiv updated 2015/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the following propositions. Theorem 1: Let M be a subfield of a fixed algebraic closure \Q of \Q whose existential elementary theory is decidable (resp. primitively decidable). Then, M is conjugate to a recursive (resp. primitive recursive) subfield L ⊂ \Q. Theorem 2: For each positive integer e there are infinitely many e-tuples \boldsymbol σ∈ \Gal(\Q)e such that the field \Q( \boldsymbol σ) -- the fixed field of \boldsymbol σ, is recursive in \Q and its elementary theory is decidable. Moreover, \Q(\boldsymbol σ) is PAC and \Gal(\Q(\boldsymbol σ)) is isomorphic to the free profinite group on e generators.