Orders in the Euclidean court

Abstract Following Johnson, Queen, and Sevilla, we call an order $$\mathcal {O}$$ O in a number field K if, for all $$\alpha ,\beta \in \mathcal {O}$$ α , β ∈ O with $$\beta \ne 0$$ β ≠ 0 for which $$\alpha \mathcal {O}+\beta \mathcal {O}$$ α O + β O is a principal ideal of $$\mathcal {O}$$ O , there are $$\gamma , \rho \in \mathcal {O}$$ γ , ρ ∈ O with $$\alpha =\beta \gamma + \rho $$ α = β γ + ρ and $$|N_{K/\mathbb {Q}}(\rho )| < |N_{K/\mathbb {Q}}(\beta )|$$ | N K / Q ( ρ ) | < | N K / Q ( β ) | . We prove that only finitely many quadratic orders are generalized Euclidean. Additionally, assuming GRH, we show that for each fixed $$d > 1,$$ d > 1 , there are only finitely many integers m for which $$T^d-m$$ T d - m is irreducible over $$\mathbb {Q}$$ Q and $$\mathbb {Z}[m^{1/d}] = \mathbb {Z}[T]/(T^d-m)$$ Z [ m 1 / d ] = Z [ <