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 the Hasse norm principle holds for $K/k$. Some examples of metacyclic, dihedral, quasidihedral, modular, generalized quoternion, extraspecial groups and $Z$-groups $G$ with trivial Schur multiplier $M(G)=0$ are given. As a consequence, we get the Tamagawa number $\tau(T)=|G<sup>{ab}|/|H<sup>{ab}|$ of the norm one tori $T=R<sup>{(1)}_{K/k}(G_m)$ of $K/k$ via Ono's formula $\tau(T)=|H<sup>1(k,\widehat{T})|/|Sha(T)|$ where $G<sup>{ab}$ and $H<sup>{ab}$ are the abelianizations of $G$ and $H$ respectively, $\widehat{T}={\rm Hom}(T,G_m)$ is the character module of $T$ and $Sha(T)$ is the Shafarevich-Tate group of $T$.