On Double Covers of the Generalized Symmetrical Group [formula omitted]d ≀ [formula omitted]m as Galois Groups over Algebraic Number Fields K with μ d ⊂ K

Abstract

Let K be an algebraic number field which contains the dth roots of unity. We will prove that all double covers of the generalized symmetric group Z d ≀ S m are realizable as a Galois group over K and over K( T), if d is odd. If d is even, we will determine all double covers of Z d ≀ G m which can be shown to be Galois groups over K and over K( T) using Serre′s formula on trace forms. If d ≠ 1 we will use trinomials ƒ( X d ) such that the Galois group of ƒ( X) = X m + aX l + b ∈ K[ X] is S m .