The Hasse norm principle for some extensions of degree having square-free prime factors

We determine the structure of the obstruction group of the Hasse norm principle for a finite separable extension $K\mathrm{/}k$ of a global field of degree $d$, where $d$ has a square-free prime factor $p$ and a $p$-Sylow subgroup of the Galois group $G$ of the Galois closure of $K\mathrm{/}k$ is normal in $G$. Specifically, we give a partial classification of the validity of the Hasse norm principle for $K\mathrm{/}k$ in the case where (1) $[K:k]=p\mathrm{\ell }$ where $p$ and $\mathrm{\ell }$ are two distinct prime numbers; or (2) $[K:k]=4p$ where $p$ is an odd prime. The result (1) gives infinitely many new existences of finite extensions of arbitrary number fields for which the Hasse norm principle fail. Furthermore, we prove that there exist infinitely many separable extensions of square-free degree for which the exponents of the obstruction groups to the Hasse norm principle are not prime powers.