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Factorial groups and Pólya groups in Galoisian extension of $$\mathbb Q$$. (English) Zbl 1107.13303
Fontana, Marco (ed.) et al., Commutative ring theory and applications. Proceedings of the fourth international conference, Fez, Morocco, June 7-12, 2001. New York, NY: Marcel Dekker (ISBN 0-8247-0855-5/pbk). Lect. Notes Pure Appl. Math. 231, 77-86 (2003).
Summary: The classical factorials in $$\mathbb{Z}$$ may be generalized by factorial ideals in every integral domain $$D$$. When $$D$$ is completely integrally closed, these factorial ideals generate a subgroup of the divisorial group, the factorial group $$\text{Fact}(D)$$, and, if $$D$$ is a Krull domain, $$\text{Fact}(D)$$ is a free abelian group. The classes of the factorial ideals generate the Pólya group $$\text{Po}(D)$$ and we give some properties of this group $$\text{Po}(D)$$, especially when $$D$$ is the ring of integers of a finite Galoisian extension of $$\mathbb{Q}$$.
For the entire collection see [Zbl 1027.00012].

##### MSC:
 13C20 Class groups 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F20 Polynomial rings and ideals; rings of integer-valued polynomials