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Arithmetic of non-principal orders in algebraic number fields

Andreas Philipp — 2010

Actes des rencontres du CIRM

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

On the Olson and the Strong Davenport constants

Oscar OrdazAndreas PhilippIrene SantosWolfgang A. Schmid — 2011

Journal de Théorie des Nombres de Bordeaux

A subset S of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of S is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, p -groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...

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