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

AbstractLet K be a real algebraic number field. Suppose that G occurs as a Galois group of a normal real extension field of K. Using elementary methods, we show that certain types of split extensions of an elementary abelian 2-group by G also occur as Galois groups of normal real extensions of K. Among other examples, we show that Sylow 2-subgroups of the symmetric and alternating groups of degree 2n, as well as the Weyl groups of type Bn and Dn, occur as Galois groups of real extensions of the rationals