Shortest division chains in imaginary quadratic number fields

Abstract

Let O _d be the set of algebraic integers in an imaginary quadratic number field Q[√d], d<0, where d is the discriminant of O _d . Consider the Euclidean Algorithm (EA), applied to algebraic integers ξ, η ∈ O _d . It consists in computing a sequence of remainders ρ ₀=ξ, ρ ₁=η, ρ ₂, ..., ρ _n+1=0, where ρ _i+1=ρ _i−1−γ _i ρ _i for algebraic integers γ _i ∈ K, i=1, ..., n. We show that except for d=−11 the number of divisions to be carried out is always minimized by choosing each γ _i such that N(ρ _i-1 - γ _i ρ _i ), the norm of γ _i-1 - γ _i ρ _i , is minimal. This result has been proven previously in special cases. It also applies to those imaginary quadratic number rings which are not Euclidean; in this case the division chains may be infinite. For d=−7, −8 the methods applied so far must be modified somewhat, and for d=−11 we provide a counterexample and a theorem which partially answers the question, how shortest division chains can be obtained.