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

TL;DR

This paper investigates the failure of the Hasse norm principle in certain non-Galois number field extensions of square-free degree, identifying specific conditions under which the principle does not hold.

Contribution

It establishes the existence of non-Galois extensions of specified degrees where the Hasse norm principle fails, using group cohomology and representation theory techniques.

Findings

Hasse norm principle fails for extensions of degree divisible by 3, 55, 91, or 95.

Determines the structure of Tate--Shafarevich groups for these extensions.

Reduces the problem to analyzing 2-dimensional representations over finite fields.

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 $p$.