Hasse norm principle for metacyclic extensions with trivial Schur multiplier
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908623618179072 |
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| author | Hoshi, Akinari Yamasaki, Aiichi |
| author_facet | Hoshi, Akinari Yamasaki, Aiichi |
| contents | 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 $M(G)=0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hasse norm principle for metacyclic extensions with trivial Schur multiplier Hoshi, Akinari Yamasaki, Aiichi Number Theory Algebraic Geometry 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 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 $M(G)=0$. |
| title | Hasse norm principle for metacyclic extensions with trivial Schur multiplier |
| topic | Number Theory Algebraic Geometry 11E72, 12F20, 13A50, 14E08, 20C10, 20G15 |
| url | https://arxiv.org/abs/2503.14365 |