Pólya fields and Pólya numbers

Amandine Leriche[1]

  • [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 (2010)

  • Volume: 2, Issue: 2, page 21-26
  • ISSN: 2105-0597

Abstract

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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 .

How to cite

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Leriche, Amandine. "Pólya fields and Pólya numbers." Actes des rencontres du CIRM 2.2 (2010): 21-26. <http://eudml.org/doc/196292>.

@article{Leriche2010,
abstract = {A number field $K$, with ring of integers $\mathcal\{O\}_K$, is said to be a Pólya field if the $\mathcal\{O\}_K$-algebra formed by the integer-valued polynomials on $\mathcal\{O\}_K$ 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 $po_K$ of a Pólya field containing $K$, namely the Pólya number of $K$.},
affiliation = {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},
author = {Leriche, Amandine},
journal = {Actes des rencontres du CIRM},
keywords = {Pólya fields; Hilbert class field; genus field; integer-valued polynomials; Pólya field; integral-valued polynomials; regular basis; Pólya extension},
language = {eng},
number = {2},
pages = {21-26},
publisher = {CIRM},
title = {Pólya fields and Pólya numbers},
url = {http://eudml.org/doc/196292},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Leriche, Amandine
TI - Pólya fields and Pólya numbers
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 21
EP - 26
AB - A number field $K$, with ring of integers $\mathcal{O}_K$, is said to be a Pólya field if the $\mathcal{O}_K$-algebra formed by the integer-valued polynomials on $\mathcal{O}_K$ 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 $po_K$ of a Pólya field containing $K$, namely the Pólya number of $K$.
LA - eng
KW - Pólya fields; Hilbert class field; genus field; integer-valued polynomials; Pólya field; integral-valued polynomials; regular basis; Pólya extension
UR - http://eudml.org/doc/196292
ER -

References

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  9. A. Leriche, Groupes, Corps et Extensions de Pólya : une question de capitulation, PhD thesis, Université de Picardie Jules Verne (Décembre 2010). 
  10. A. Leriche, Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, Journal de Théorie des Nombres de Bordeaux, 23, (2011), 235-249. Zbl1282.13040MR2780627
  11. A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpren, J. Reine Angew. Math.149 (1919), 117-124. Zbl47.0163.05
  12. G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math.149 (1919), 97-116. Zbl47.0163.04
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  14. H. Zantema, Integer valued polynomials over a number field, Manusc. Math.40 (1982), 155-203. Zbl0505.12003MR683038

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