Computation of the Euclidean minimum of algebraic number fields

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri ([7]). With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields of degree up to 8 in any signature, especially for cyclotomic fields, and many new examples of norm-Euclidean or non-norm-Euclidean algebraic number fields. Then, we show how to apply the algorithm to study extensions of norm-Euclideanity. We consider an algebraic number field K. Let ZK be its ring of integers. We write r1 for its number of real places, 2r2 for its number of imaginary places and n = r1 + 2r2 for its degree. We denote by NK/Q the usual norm. The pair (r1, r2) is called the signature of K. We write d(K) for the discriminant of K, hK for the class number of K, ZK for the group of units of K and r = r1 + r2 − 1 for its rank. Definition (Euclideanity with respect to the norm). We say that ZK is Euclidean with respect to the norm if and only if for every (a, b) ∈ ZK ×ZK \{0}, there exists some c ∈ ZK such that ∣NK/Q(a− bc) ∣∣ < ∣NK/Q(b) ∣∣ . If the property written above holds, we also say that K is norm-Euclidean or that NK/Q is an Euclidean algorithm for K. There is no reason to choose the norm instead of another Euclidean algorithm, but the multiplicative property of the norm makes it (relatively) easier to test if NK/Q is an Euclidean algorithm for ZK . Indeed, checking if ZK is norm-Euclidean is equivalent to checking if for any ξ ∈ K, there exists some z ∈ ZK such that ∣NK/Q(ξ − z) ∣∣ < 1. Therefore, the determination of norm-Euclideanity can be seen in a geometric setting. The notion of Euclidean minimum will be introduced to indicate the “distance” between K and the lattice ZK . This notion of Euclideanity was extensively studied for several purposes. First, the existence of an Euclidean algorithm provides a technique to compute greatest common divisors in ZK . Besides, if ZK is Euclidean, then it is a principal ideal domain and therefore a unique factorisation domain. Consequently, in the 19 century, Wantzel tried to fill one gap in Lamé’s “proof” of Fermat’s Last Theorem by using some properties of norm-Euclideanity. Following and correcting his ideas, Cauchy and Kummer studied cyclotomic fields and proved that some of them are norm-Euclidean (see [15] for both mathematical and historical details). Date: 2nd October 2012. 2010 Mathematics Subject Classification. Primary 11Y40; Secondary 11R04, 11A05, 13F07.