Orders in the Euclidean Court

Following Johnson, Queen, and Sevilla, we call an order $\mathcal{O}$ in a number field $K$ $\mathsf{\text{generalized}}\mathsf{\text{Euclidean}}$ if for all $\alpha ,\beta \in \mathcal{O}$ with $\beta \mathrm{\ne }0$ for which $\alpha \mathcal{O}+\beta \mathcal{O}$ is a principal ideal of $\mathcal{O}$, there are $\gamma ,\rho \in \mathcal{O}$ with $\alpha =\beta \gamma +\rho$ and $\mathrm{\mid }{N}_{K\mathrm{/}\mathbb{Q}}(\rho )\mathrm{\mid }<\mathrm{\mid }{N}_{K\mathrm{/}\mathbb{Q}}(\beta )\mathrm{\mid }$. We prove that only finitely many quadratic orders are generalized Euclidean. Additionally, assuming GRH, we show that for each fixed $d>1$ there are only finitely many integers $m$ for which $T^d-m$ is irreducible over $\mathbb{Q}$ and $\mathbb{Z}[m^{1\mathrm{/}d}]=\mathbb{Z}[T]\mathrm{/}(T^d-m)$ is generalized Euclidean. When $d$ is even, we can remove the assumption of GRH.