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If a and b are two integers, with b 6= 0, then there exist two integers q
and r such that a = bq + r, and that |r| < |b|. This so-called Euclidean division
property plays a fundamental role in the arithmetic of the usual integers. It is
natural to try to extend this to more general rings, for instance rings of integers of
algebraic number fields. This idea leads to the notions of Euclidean number fields
and Euclidean minima. Both are very classical topics of number theory. The aim
of this talk is to survey old and new results concerning this subject, such as new
Euclidean number fields and upper bounds for Euclidean minima. In particular, we
will survey the history and recent developments concerning a classical conjecture of
Minkowski.
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