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Section: New Results

Non commutative number theory

Participant : Jean Paul Cerri.

Pierre Lezowski has studied in [12], Euclidean properties of matrix algebras. He proved that if A is a commutative ring and if n>1 is an integer , then Mn(A) is right and left Euclidean if and only if A is a principal ideal ring. Moreover, under the hypothesis that the stathm takes integer values, he established that if A is an integral domain, then Mn(A) is ω-stage right and left Euclidean if and only if A is a Bézout ring. He also proved, under the same hypothesis, that if A is a K-Hermite ring, then Mn(A) is (4n-3)-stage left and right Euclidean, that if A is an elementary divisor ring, then Mn(A) is (2n-1)-stage left and right Euclidean, and that if A is a principal ideal ring, then Mn(A) is 2-stage right and left Euclidean. In each case, he obtained an explicit algorithm allowing to compute, among other things, right or left gcd in Mn(A).

Jean-Paul Cerri and Pierre Lezowski have generalized in [19], Cerri's algorithm (for the computation of the upper part of the norm-Euclidean spectrum of a number field) to totally definite quaternion fields. This allowed them to establish the exact value of the norm-Euclidean minimum of many orders in totally definite quaternion fields over a quadratic number field. Before this work, nobody knew how to compute the exact value of such a minimum when the base number field has degree >1. They also proved that the Euclidean minimum and the inhomogeneous minimum of orders in such quaternion fields are always equal and that moreover they are rational under the hypothesis that the base number field is not quadratic, which remains the only open case, as for real number fields.

In [13] Lezowski determines which cyclic field of degree d are norm-Euclidean for

d=5,7,19,31,37,43,47,59,67,71,73,79,97.