Diophantine definability over non-finitely generated non-degenerate modules of algebraic extensions of \(\mathbb{Q}\) (Q5944053)

One of the most famous theorems of the last century was the solution to Hilbert's 10th problem by Matiyasevich, building on work of Putnam, Davis, Robinson etc. Naturally, one can ask what computable rings admit a positive/negative solution to this problem. Of central interest are infinite algebraic extensions of the rationals [see, e.g., \textit{A. Shlapentokh}, Commun. Pure Appl. Math. 42, 939-962 (1989; Zbl 0695.12020)]. The author investigates non-degenerate modules contained in infinite algebraic extensions of the rationals. The central issues are concerned with Diophantine definability. The methods of this positive result are, as one would expect, algebraic.