Abstract
We show that elliptic curves whose Mordell–Weil groups are finitely generated over some infinite extensions of \({\mathbb {Q}}\) , can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.
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Shlapentokh, A. Rings of algebraic numbers in infinite extensions of \({\mathbb {Q}}\) and elliptic curves retaining their rank. Arch. Math. Logic 48, 77–114 (2009). https://doi.org/10.1007/s00153-008-0118-y
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DOI: https://doi.org/10.1007/s00153-008-0118-y