Submission date: 27 September 2013.

Abstract:

In this note we give a negative answer to Abraham Robinson's question whether a finitely generated extension of an undecidable field is always undecidable. We construct 'natural' undecidable fields of transcendence degree 1 over Q all of whose proper finite extensions are decidable. We also construct undecidable algebraic extensions of Q that allow decidable finite extensions.

Mathematics Subject Classification: 12L05, 12E30

Keywords and phrases:

Full text arXiv 1309.7138: pdf, ps.