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.

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri in 2007. With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields of degree up to 8 8 in any signature, especially for cyclotomic fields, and many new examples of norm-Euclidean or non-norm-Euclidean algebraic number fields. Then, we show how to apply the algorithm to study extensions of norm-Euclideanity.

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In [11], Lenstra defined the notion of Euclidean ideal class. Using a slight modification of an algorithm described in [12], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari ([5]), we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class.