Abstract
LetR be an integral domain whose quotient field is an algebraic number field. Cooke and Weinberger [4] showed that, assuming the Generalized Riemann Hypothesis, ifR is a principal ideal domain and has infinite unit group, thenR is 4-stage Euclidean with the absolute value of the norm as algorithm. We remove the assumption of the Generalized Riemann Hypothesis from this result for totally real Galois extensions of ℚ of degree greater than or equal to three, replacing it with the requirement of finding sufficiently many prime elements ofR, ℚ such that the unit group ofR maps onto (R/((π1⋯πr)2))* via the reduction map. A similar result holds for real quadratic fields.
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