Diophantine definability over non-finitely generated non-degenerate modules of algebraic extensions of ℚ

Abstract.

We investigate the issues of Diophantine definability over the non-finitely generated version of non-degenerate modules contained in the infinite algebraic extensions of the rational numbers. In particular, we show the following. Let k be a number field and let K _inf be a normal algebraic, possibly infinite, extension of k such that k has a normal extension L linearly disjoint from K _inf over k. Assume L is totally real and K _inf is totally complex. Let M _inf be a non-degenerate O _k -module, possibly non-finitely generated and contained in O _Kinf . Then M _inf contains a submodule M¯ _inf such that M _inf /M¯ _inf is torsion and O _k has a Diophantine definition over M¯ _inf .