Pólya fields and Pólya numbers

Leriche, Amandine

doi:10.5802/acirm.29

Amandine Leriche

Actes des rencontres du CIRM, Volume 2 (2010) no. 2, p. 21-26

Abstract

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 $K$ .

Published online : 2011-10-20
DOI : https://doi.org/10.5802/acirm.29
Classification: 11R04, 13F20, 11R16, 11R37
Keywords: Pólya fields, Hilbert class field, genus field, integer-valued polynomials

@article{ACIRM_2010__2_2_21_0,
     author = {Amandine Leriche},
     title = {P\'olya fields and P\'olya numbers},
     journal = {Actes des rencontres du CIRM},
     publisher = {CIRM},
     volume = {2},
     number = {2},
     year = {2010},
     pages = {21-26},
     doi = {10.5802/acirm.29},
     language = {en},
     url = {https://acirm.centre-mersenne.org/item/ACIRM_2010__2_2_21_0}
}

Leriche, Amandine. Pólya fields and Pólya numbers. Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 21-26. doi : 10.5802/acirm.29. https://acirm.centre-mersenne.org/item/ACIRM_2010__2_2_21_0/

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