Hasse norm principle for metacyclic extensions with trivial Schur multiplier

Abstract

Let $k$ be a global field, $K/k$ be a finite separable field extension and $L/k$ be the Galois closure of $K/k$ with Galois groups $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$. In 1931, Hasse proved that if $G$ is cyclic, then the Hasse norm principle holds for $K/k$. We show that if $G$ is metacyclic with trivial Schur multiplier $M(G)=0$, then $H$ is cyclic and the Hasse norm principle holds for $K/k$. Some examples of metacyclic, dihedral, quasidihedral, modular, generalized quaternion, extraspecial groups and $Z$-groups $G$ with trivial Schur multiplier $M(G)=0$ are given. These provide new examples which the Hasse norm principle hold for non-Galois extensions $K/k$ whose Galois closure is $L/k$ with metacyclic $G={\rm Gal}(L/k)$ and $M(G)=0$.