On the Northcott property and other properties related to polynomial mappings (Q2841494)

Let \(K\) be a subfield of the field of all algebraic numbers and for \(d>1\) let \(K^{(d)}\) be the composite of all extensions of \(K\) with degree \(\leq d\), put \(K^{(d)^{1}}=K^{(d)}\), and for \(n=1,2,\dots\) put \(K^{(d)^{n+1}}=\left(K^{(d)^{n}}\right)^{(d)}\). The field \(K\) is said to have the Northcott property if it contains only finitely many numbers having bounded absolute logarithmic Weil height, and it has the Bogomolov property if there exists \(T>0\) such that the only elements of \(K\) with height smaller than \(T\) are roots of unity.NEWLINENEWLINEThe authors show that if the extension \(K/\mathbb Q\) is Galois, then \(K^{(d)}/\mathbb Q\) is also Galois, and if the Galois group of \(K/\mathbb Q\) has finite exponent \(b\), then the group of \(K^{(d)}/\mathbb Q\) has exponent bounded by \(d!b\). As a consequence they obtain that the maximal Abelian subfield \(K^{(d)}_{Ab}\) of \(K^{(d)}\) has the Northcott property (for \(K=\mathbb Q\) this has been earlier established by \textit{E. Bombieri} and \textit{U. Zannier} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 12, No. 1, 5--14 (2001; Zbl 1072.11077)], and if the Galois group of \(K/\mathbb Q\) has finite exponent, then for any \(d,n\) the field \(K^{(d)^{n}}\) has the Bogomolov poperty.NEWLINENEWLINEIn the next sections it is shown that there exist fields with Northcott property which are not contained in \(\mathbb Q^{(d)}\) for any \(d\) and have uniformly bounded local degrees, and also subfields of \(\mathbb Q^{d}\) (for certain \(d\)) with Northcott property whose subfields of finite degree do not have uniformly bounded degrees over \(\mathbb Q\).NEWLINENEWLINEIn the final section the authors consider polynomial mappings and show that their results imply answers to certain old questions listed by the reviewer in [Polynomial mappings. Lecture Notes in Mathematics. 1600. Berlin: Springer-Verlag. (1995; Zbl 0829.11002)].