On cogalois extensions

Given a number field F and a finite abelian group View the MathML source, it is proven that there exists an extension K/F which is Galois and cogalois with Gal(K/F) ? cog(K/F) ? G iff the primitive n11-th roots of unity are present in F and the field obtained by adjoining the n1-th roots of unity to F is pure over F.