Euclidean totally definite quaternion fields over the rational field and over quadratic number fields

Jean-Paul Cerri (1, 2) , Jérôme Chaubert (3) , Pierre Lezowski (1, 2)

1 IMB - Institut de Mathématiques de Bordeaux
2 LFANT - Lithe and fast algorithmic number theory
3 EPFL - Ecole Polytechnique Fédérale de Lausanne

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.

Mots clés: Euclidean order; norm-Euclidean order.; norm-Euclidean order; Quaternion field; Euclidean algorithm

Domaines: Théorie des nombres [math.NT]

Jean-Paul Cerri, Jérôme Chaubert, Pierre Lezowski. Euclidean totally definite quaternion fields over the rational field and over quadratic number fields. International Journal of Number Theory, 2013, 9 (3), pp.653-673. ⟨10.1142/S1793042112501540⟩. ⟨hal-00738164v2⟩