On a question of Abraham Robinson's

17 October 2013
Jochen Koenigsmann
<div>We give a negative answer to Abraham Robinson's question whether a finitely generated extension of an undecidable field is always undecidable by constructing undecidable fields of transcendence degree 1 over the rationals all of whose proper finite extensions are decidable. We also construct undecidable algebraic extensions of the rationals which allow decidable finite extensions.</div>