Orders in the Euclidean Court
Following Johnson, Queen, and Sevilla, we call an order $\mathcal{O}$ in a number field $K$ \textsf{generalized Euclidean} if for all $α,β\in \mathcal{O}$ with $β\ne 0$ for which $α\mathcal{O}+β\mathcal{O}$ is a principal ideal of $\mathcal{O}$, there are $γ, ρ\in \mathcal{O}$ with $α=βγ+ ρ$ and $|N_{K/\mathbb{Q}}(ρ)| < |N_{K/\mathbb{Q}}(β)|$. 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 $\mathbb{Z}[m^{1/d}] = \mathbb{Z}[T]/(T^d-m)$ is generalized Euclidean. When $d$ is even, we can remove the assumption of GRH.
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