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This article deals with the determination of the Euclidean minimum M(K) of a totally real number field K of degree n ≥ 2, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree 2 to 8 and small discriminants, most of which were previously unknown. Tables are given at… (More)
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 give examples of Generalized Euclidean but not norm-Euclidean number fields of degree greater than 2. In the same way we give examples of 2-stage Euclidean but not norm-Euclidean number fields of degree greater than 2. In both cases, no such examples were known.
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.