A note on the Northcott property and undecidability (Q2788653)

The authors show a natural relationship between the Julia Robinson number (JR number) (which in the case of the ring of the totally real algebraic integers was equal 4) and the Norhcott property of a ring of totally real algebraic integers. They also show a relationship between the JR number and the norm of a ring of totally algebraic integers. As a consequence they show that any subring of a ring of totally real integers with the Northcott property has an undecidable first-order theory. This also implies that the compositum of all totally real abelian extensions of \(\mathbb{Q}\) of bounded degree \(d\) has undecidable first-order theory. They also provide an explicit construction for \(d = 3\).