For any quadratic extension K/F, there exists an abelian variety over F of positive rank whose rank is unchanged after base change to K, yielding undecidability of integer polynomial equations over every number field.
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Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field
For any quadratic extension K/F, there exists an abelian variety over F of positive rank whose rank is unchanged after base change to K, yielding undecidability of integer polynomial equations over every number field.