Two remarks on Narkiewicz's property (P)

Abstract

Due to Narkiewicz a field $F$ has property (P) if for no polynomial $f\in F[x]$ of degree at least two there is an infinite $f$-invariant subset of $F$. We present a new example of an algebraic extension of $\mathbb{Q}$ satisfying (P). This is the first example in which we can find points of arbitrarily small positive Weil-height. Moreover, we study the possibility of property (P) for the field generated by all symmetric Galois extensions of $\mathbb{Q}$. In particular we prove that there are no infinite backward orbits of non linear polynomials in this field.