Totally indefinite Euclidean quaternion fields
Volume 165 / 2014
Acta Arithmetica 165 (2014), 181-200 MSC: Primary 13F07; Secondary 11Y40, 16H05, 16H10. DOI: 10.4064/aa165-2-4
We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a 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.