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arXiv · 2307.12550

The Hasse norm principle for some non-Galois extensions of square-free degree

Yasuhiro Oki
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Abstract

In this paper, we study the Hasse norm principle for some non-Galois extensions of number fields. Our main theorem is that for any square-free composite number $d$ which is divisible by at least one of $3$, $55$, $91$ or $95$, there exists a finite extension of degree $d$ for which the Hasse norm principle fails. To accomplish it, we determine the structure of the Tate--Shafarevich groups of norm one tori for finite extensions of degree $d$ under the normality of $p$-Sylow subgroups of the Galois groups of their Galois closures for a square-free prime factor $p$ of $d$. Moreover, we reduce the assertion to an investigation of $2$-dimensional $\mathbb{F}_p$-representations of some groups of order coprime to $p$.

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Yasuhiro Oki. 2023-07-24. The Hasse norm principle for some non-Galois extensions of square-free degree. https://arxiv.org/abs/2307.12550

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