Arithmetic in quadratic fields with unique factorization

Abstract

In a quadratic field Q($\sqrt D$), 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. We prove that for D⩽−19 even remainder sequences with possibly non-decreasing norms cannot determine the GCD of arbitrary inputs. We then show how to compute the greatest common divisor of the algebraic integers in any fixed Q($\sqrt D$) with class number 1 in O (S ²) binary steps where S is the number of bits needed to encode the inputs. We also prove that in any domain the computation of the prime factorization of an algebraic integer can be reduced in polynomial-time to factoring its norm into rational primes. Our reduction is based on a constructive version of a theorem by A. Thue. Finally we present another GCD algorithm for complex quadratic fields based on a short lattice vector construction.