Shortest division chains in imaginaryquadratic 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 ∈ O d , 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 ρ 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.