Pólya fields and Pólya numbers

Leriche, Amandine

Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 21-26.

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 $K$.

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

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