Orders in the Euclidean court

Following Johnson, Queen, and Sevilla, we call an order O O in a number field K if, for all α ,β ∈ O α , β ∈ O with β ≠ 0 β ≠ 0 for which α O+β O α O + β O is a principal ideal of O O , there are γ , ρ ∈ O γ , ρ ∈ O with α =β γ + ρ α = β γ + ρ and |NK/Q(ρ )| < |NK/Q(β )| | 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 Tᵈ-m T d - m is irreducible over Q Q and Z[m1/d] = Z[T]/(Tᵈ-m) Z [ m 1 / d ] = Z [ <