On the properties of Northcott and of Narkiewicz for fields of algebraic numbers

Abstract

This paper surveys on some recent results concerning certain finiteness properties for subfields $K$ of $\overline{\mathbb{Q}}$: first, the so-called \textit{Northcott property} of finiteness of elements in $K$ of bounded Weil height and then the \textit{Property (P)} of finiteness of possible subsets of $K$ sent onto themselves by some polynomial of degree $>1$. The first was established by Northcott for the union of the fields of given degree over $\mathbb{Q}$; the second one was introduced by Narkiewicz; it is also related to preperiodic points for polynomial maps. It is known that the first implies the second, so they both hold for number fields. As to fields of infinite degree over $\mathbb{Q}$, we shall see some criteria for the first property, and hence for the second, but we shall also see that the second property does not imply the first. Some of these constructions provide answers, both in the positive and in the negative as the case may be, to questions explicitly formulated by Narkiewicz.