We show that for any quadratic extension of number fields $K/F$ , there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$ . This result implies that Hilbert’s tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring ${\mathcal{O}}_{K}$ of integers of any number field $K$ , there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over ${\mathcal{O}}_{K}$ has solutions in ${\mathcal{O}}_{K}$ .
