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.

We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish the complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.

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