Construction of some wreath products as Galois groups of normal real extensions of the rationals (Q1079599)

Let \(K\) be a real number field. Let \(f(X)\in K[X]\) be a real irreducible polynomial of degree \(n\) with Galois group \(G\) over \(K\). The author proves that there exists a rational number \(c\) such that the polynomial \(g(X)=f(X^ 2+c)\) is real with Galois group over \(K\) isomorphic to the wreath product of \({\mathbb Z}/2{\mathbb Z}\) by \(G\) of order {\#}G\(\cdot 2^ n\). Using this result he proves that a Sylow 2-subgroup of \(S_{2^ n}\), of \(A_{2^ n}\), and of \(\text{GL}(n,2)\), as well as the Weyl groups of type \(A_ n\), \(B_ n\) and \(D_ n\), occurs as a Galois group of a real normal extension of \({\mathbb Q}\). Remark that since this wreath product of \(G\) is a split extension of \(G\) with abelian kernel, it is known that it appears as a Galois group over \({\mathbb Q}\) [cf. \textit{J. Neukirch}, Invent. Math. 21, 59--116 (1973; Zbl 0267.12005)].