Height lower bounds for elements of highly composite rings

Let $\mathbb{Q}^{(d)}$ be the composite field of all number fields of degree at most $d$. In 2001 Bombieri and Zannier proved that $\mathbb{Q}^{(2)}$ has the Northcott property and asked what happens for $d\geq 3$. Here we study the absolute Weil height for elements in the composite ring of the rings of integers of such number fields. In particular, we consider $\mathbb{Q}^{(3)}$ as the composite field of $\mathbb{Q}^{(2)}$ and a minimal infinite family of cubic fields, and we show that the composite ring of the rings of integers of these fields does have the Northcott property. Our results follow from new height lower bounds, expressed in terms of the degree. Moreover, we introduce a notion of size for subfields of $\mathbb{Q}^{(3)}$. For instance, $\mathbb{Q}^{(3)}$ has size $1$ and the maximal abelian subfield of $\mathbb{Q}^{(3)}$ has size $1/2$. We show that there is a subfield of $\mathbb{Q}^{(3)}$ of size $1$ which has the Northcott property.