Totally indefinite Euclidean quaternion fields

Abstract : 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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Jean-Paul Cerri, Jérôme Chaubert, Pierre Lezowski. Totally indefinite Euclidean quaternion fields. Acta Arithmetica, Instytut Matematyczny PAN, 2014, 165 (2), pp.181-200. 〈hal-01016614v3〉

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