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.
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© 1989 Springer-Verlag Berlin Heidelberg
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Rolletschek, H. (1989). Shortest division chains in imaginary quadratic number fields. In: Gianni, P. (eds) Symbolic and Algebraic Computation. ISSAC 1988. Lecture Notes in Computer Science, vol 358. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51084-2_21
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DOI: https://doi.org/10.1007/3-540-51084-2_21
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