Non-Galois cubic fields which are Euclidean but not norm-Euclidean

Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm–Euclidean. Let R be the ring of integers of an algebraic number field K. A Euclidean algorithm on R is a map φ : R → N such that φ(r) 6= 0 for r 6= 0, and for all a, b ∈ R, b 6= 0, there exist q, r ∈ R with a = qb + r and φ(r) < φ(b). If φ is completely multiplicative, that is, φ(ab) = φ(a)φ(b), then φ can be extended to a completely multiplicative function on K. The Euclidean property can be expressed as follows: for every x ∈ K there is γ ∈ R such that φ(x − γ) < 1. The problem which has been studied most often is the determination of those number fields for which the absolute value of the norm is a Euclidean algorithm. We refer to this function simply as the norm and denote it by N . We call a field norm–Euclidean when the norm is a Euclidean algorithm for the field. Weinberger [9] showed that, under the assumption of the Generalized Riemann Hypothesis for Dedekind zeta functions, if there are infinitely many units in R, then R is a Euclidean domain if and only if it is a principal ideal domain. The assumption of the Generalized Riemann Hypothesis was removed in [2] and [4] for totally real Galois extensions K of Q with degree greater than or equal to 3, with the requirement to find sufficiently many nonassociate prime elements π1, . . . , πn of R such that the unit group of the ring of integers maps onto (R/(π1π2 · · ·πn)) via the reduction map. In [3], the ring of integers of Q( √ 69) was shown to be Euclidean but not norm–Euclidean. This paper may be viewed as an extension of the ideas in [3] to cubic fields. We will show that the cubic fields with discriminants −327 and 1929 are Euclidean but not norm–Euclidean. Taylor [8] and Smith [7], respectively, showed that these fields are not norm–Euclidean. We outline the method for defining our Euclidean algorithm. First, we determine the set B of elements modulo which there exists a coprime residue class which does not contain elements of smaller norm. Equivalently, we determine the elements x of K such that minγ∈RN(x − γ) ≥ 1. Now we try to define a new completely multiplicative Euclidean algorithm on the ring of integers by setting it equal to the norm for primes not dividing elements of B and increasing the value at primes which do divide the elements of B. Received by the editor February 18, 1994 and, in revised form, April 15, 1994, August 11, 1994, and February 22, 1995. 1991 Mathematics Subject Classification. Primary 11A05; Secondary 11R16. c ©1996 American Mathematical Society