Pólya fields and Pólya numbers

Leriche, Amandine¹

¹ LAMFA, CNRS UMR 6140 Université de Picardie Jules Verne 33, rue Saint-Leu 80039 Amiens & École Centrale de Lille Cité Scientifique 59650 Villeneuve d’Ascq France

Actes des rencontres du CIRM, Troisième Rencontre Internationale sur les Polynômes à Valeurs Entières, Tome 2 (2010) no. 2, pp. 21-26

Résumé

A number field $K$, with ring of integers ${𝒪}_K$, is said to be a Pólya field if the ${𝒪}_K$-algebra formed by the integer-valued polynomials on ${𝒪}_K$ admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field $K$ is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of $K$ in a Pólya field. We give a positive answer to this embedding problem by showing that the Hilbert class field $H_K$ of $K$ is a Pólya field. Finally, we give upper bounds for the minimal degree $p{o}_K$ of a Pólya field containing $K$, namely the Pólya number of $K$.

Détail

Publié le : 2011-10-20
DOI : 10.5802/acirm.29

Classification : 11R04, 13F20, 11R16, 11R37
Keywords: Pólya fields, Hilbert class field, genus field, integer-valued polynomials