Rings of algebraic numbers in infinite extensions of $${\mathbb {Q}}$$ and elliptic curves retaining their rank

Archive for Mathematical Logic 48 (1):77-114 (2009)
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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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10.1007/s00153-008-0118-y

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Defining integers.Alexandra Shlapentokh - 2011 - Bulletin of Symbolic Logic 17 (2):230-251.

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Questions of decidability and undecidability in number theory.B. Mazur - 1994 - Journal of Symbolic Logic 59 (2):353-371.

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