On cogalois extensions (Q1181430)

The authors prove the following result: Given a number field \(F\) and a finite abelian group \(G\cong\oplus^ t_{i=1}(\mathbb{Z}/n_ i\mathbb{Z})\), \(n_ 1| n_ 2|\ldots| n_ t\); then there exists an extension \(K/F\) which is Galois and co-Galois with Galois group and co- Galois group isomorphic to \(G\) if and only if the primitive \(n_{t-1}\)- th roots of unity are contained in \(F\) and the field obtained by adjoining the \(n_ t\)-th roots of unity to \(F\) is pure over \(F\). The concept of co-Galois extension was introduced by \textit{C. Greither} and \textit{D. K. Harrison} in connection with their study of radical extensions [J. Pure Appl. Algebra 43, 257-270 (1986; Zbl 0607.12015)].