On double covers of the generalized symmetric group \(\mathbb{Z}_ d\wr {\mathcal S}_ m\) as Galois groups over algebraic number fields \(K\) with \(\mu_ d\subset K\)
Let \(K\) be an algebraic number field which contains the \(d\)-th roots of unity. Let \(\mathbb{Z}_ d \wr{\mathfrak S}_ m\) be the generalized symmetric group, namely the wreath product of \(\mathbb{Z}_ d\) and the symmetric group \({\mathfrak S}_ m\). In this paper it is proved that the double covers \((\mathbb{Z}_ d\wr {\mathfrak S}_ m)^ +\), \((\mathbb{Z}_ d\wr {\mathfrak S}_ m)^ -\) and \((\mathbb{Z}_ d\wr {\mathfrak S}_ m)^ 0\) occur as Galois groups over \(K(T)\) and over \(K\), for all \(m\in \mathbb{N}\). The methods used to prove these results are similar to those used by the reviewer in [J. Algebra 116, 251-260 (1988; Zbl 0662.12011)]. Let \(f(X)\in K[X]\) be an irreducible and separable polynomial with Galois group \(G\). The Galois group of \(f(X^ d)\) which is a subgroup of \(\mathbb{Z}_ d\wr G\) is studied. A criterion for the determination of the Galois group of \(f(X^ d)\) in the case \(G={\mathfrak S}_ m\), \(m\geq 2\), is given. If \(d\neq 1\), the author considers trinomials \(f(X^ d)\) such that the Galois group of \(f(X)= X^ m+ aX^ l+ b\in K[X]\) is \({\mathfrak S}_ m\). He examines the obstruction to related Galois embedding problems, from Serre's trace formula, by using the author's previous computations of the Hasse-Witt invariant of the trace form associated to trinomials [J. Algebra 155, 211-220 (1993; Zbl 0777.11010)]. He gives effective conditions in order to have, in each case, that the Galois group of \(f(X^ d)\) is isomorphic to \(\mathbb{Z}_ d\wr {\mathfrak S}_ m\) and the obstruction to the associated Galois embedding problem is trivial.
scientific article; zbMATH DE number 554214