ABSTRACT
The classical factorials in 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 https://www.w3.org/1998/Math/MathML"> F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq658.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> act(D), and, if D is a Krull domain, https://www.w3.org/1998/Math/MathML"> F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq659.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> act(D) is a free abelian group. The classes of the factorial ideals generate the Polya group https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq660.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> o(D) and we give some properties of this group https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq661.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> o(D) especially when D is the ring of integers of a finite Galoisian extension of ℚ.
