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Computing greatest common divisors and factorizations in quadratic number fields

In a quadratic number field Q(√D), D a squarefree integer, with class number 1, any algebraic integer can be decomposed uniquely into primes, but for only 21 domains Euclidean algorithms are known. It was shown by Cohn that for D≤−19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs. We extend this result by showing that there does not even exist an input in these domains for which the GCD computation becomes possible by allowing nondecreasing norms or remainders whose norms are not as small as possible. We then provide two algorithms for computing the GCD of algebraic integers in quadratic number fields Q(√D)