This article deals with the determination of the Euclidean minimum M (K) of a totally real number field K of degree n ≥ 2, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree 2 to 8 and small discriminants, most of which were previously unknown. Tables are given at the end of this paper.

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totally definite quaternion fields over the rational field and over quadratic number fields. International Journal of Number Theory, World Scientific Publishing, 2013, 9 (3), pp.653-673. <10.1142/S1793042112501540>. EUCLIDEAN TOTALLY DEFINITE QUATERNION FIELDS OVER THE RATIONAL FIELD AND OVER QUADRATIC NUMBER FIELDSJEAN-PAUL CERRI, JÉRÔME CHAUBERT, AND PIERRE LEZOWSKI Abstract. In this article we study totally definite quaternion fields over the rational field and over quadratic number fields. We establish a complete list of all such fields which are Euclidean. Moreover, we prove that every field in this list is in fact norm-Euclidean. The proofs are both theoretical and algorithmic.

Abstract. The notion of Euclidean minimum of a number field is a classical one. In this paper we generalize it to central division algebras and establish some general results in this new context.