A note on central class fields
Let K be any finite Galois extension over \({\mathbb{Q}}\). The central class field \^K of K is then defined as the maximal field such that \^K is a Galois extension over \({\mathbb{Q}}\), Gal(\^K/K) is contained in the center of Gal(\^K/ \({\mathbb{Q}})\), and \^K is unramified over K in the narrow sense. Assume that Gal(K/ \({\mathbb{Q}})\cong ({\mathbb{Z}}/\ell {\mathbb{Z}})^ 2\) with \(\ell\) a prime number. In this paper, the group structure of Gal(\^K/ \({\mathbb{Q}})\) is determined. Furthermore, a certain unramified abelian \(\ell\)-extension of a cyclic field of degree \(\ell\) is considered in connection with central class fields such as the above \^K.
scientific article; zbMATH DE number 3991626