On the properties of Northcott and of Narkiewicz for fields of algebraic numbers (Q1038633)

A subfield \(F\) of \(\overline {\mathbb Q}\), the algebraic closure of the rational field, is said to have the Northcott property, if for every \(B>0\) it contains only finitely many elements \(x\) with \(h(x)\leq B\), \(h(x)\) being Weil's logarithmic height [\textit{D. G. Northcott}, Ann. Math. (2) 51, 167--177 (1950; Zbl 0036.30102)]. The field \(F\) has the property \((P)\), if every subset \(A\subset F\) for which there exists a nonlinear polynomial \(f\in F(X)\) with \(f(A)=A\) must be finite [the reviewer, Acta Arith. 7, 241--249, ibid. 8, 11--19 (1962; Zbl 0125.00901)]. The authors give a survey of recent results dealing with these properties and prove that these two properties are not equivalent.