  Let $X$ be a projective variety over a number field $K$ endowed with a height
function associated to an ample line bundle on $X$. Given an algebraic
extension $F$ of $K$ with a sufficiently big Northcott number, we can show that
there are finitely many cycles in $X_{\bar{\mathbb{Q}}}$ of bounded degree
defined over $F$. Fields $F$ with the required properties were explicitly
constructed in arXiv:2107.09027 and arXiv:2204.04446, motivating our
investigation. We point out explicit specializations to canonical heights
associated to abelian varieties and selfmaps of $\mathbb{P}^n$. We apply
similar methods to the study of CM-points. As a crucial tool, we introduce a
refinement of Northcott's theorem.
