On quadratic subfields of generalized quaternion extensions

Аннотация

We give necessary and sufficient conditions for the embeddingof a quadratic extension of a number field $k$ into an extensionwith group of generalized quaternions; in this case, the caseof both a cyclic kernel and a generalized quaternion is considered.As a consequence, it is proved that the class of ultrasolvable$2$-extensions with cyclic kernel does not coincide with the classof non-semidirect extensions. Sufficient conditions are also givenfor the embedding of quadratic extensions $k(\sqrt{d_1})/k$,$k(\sqrt{d_2})/k$, $k(\sqrt{d_1d_2})/k$ of a number field $k$into a generalized quaternion extension $L/k$. Related examples are given.