Pólya fields, Pólya groups and Pólya extensions: a question of capitulation

Amandine Leriche[1]

  • [1] LAMFA, CNRS UMR 6140 Université de Picardie Jules Verne 33, rue Saint-Leu 80039 Amiens, France

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 1, page 235-249
  • ISSN: 1246-7405

Abstract

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A number field K , with ring of integers 𝒪 K , is said to be a Pólya field when the 𝒪 K -algebra formed by the integer-valued polynomials on 𝒪 K admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when K is not a Pólya field, we are interested in the embedding of K in a Pólya field. We study here two notions which can be considered as measures of the obstruction for K to be a Pólya field: the Pólya extensions L / K where the characteristic ideals of K extended to L become principal, and the Pólya group which is the subgroup of the class group generated by the classes of the characteristic ideals.

How to cite

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Leriche, Amandine. "Pólya fields, Pólya groups and Pólya extensions: a question of capitulation." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 235-249. <http://eudml.org/doc/219724>.

@article{Leriche2011,
abstract = {A number field $K$, with ring of integers $\mathcal\{O\}_K$, is said to be a Pólya field when the $\mathcal\{O\}_K$-algebra formed by the integer-valued polynomials on $\mathcal\{O\}_K$ admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when $K$ is not a Pólya field, we are interested in the embedding of $K$ in a Pólya field. We study here two notions which can be considered as measures of the obstruction for $K$ to be a Pólya field: the Pólya extensions $L/K$ where the characteristic ideals of $K$ extended to $L$ become principal, and the Pólya group which is the subgroup of the class group generated by the classes of the characteristic ideals.},
affiliation = {LAMFA, CNRS UMR 6140 Université de Picardie Jules Verne 33, rue Saint-Leu 80039 Amiens, France},
author = {Leriche, Amandine},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Pólya field; integral-valued polynomials; regular basis; Pólya extension},
language = {eng},
month = {3},
number = {1},
pages = {235-249},
publisher = {Société Arithmétique de Bordeaux},
title = {Pólya fields, Pólya groups and Pólya extensions: a question of capitulation},
url = {http://eudml.org/doc/219724},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Leriche, Amandine
TI - Pólya fields, Pólya groups and Pólya extensions: a question of capitulation
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 235
EP - 249
AB - A number field $K$, with ring of integers $\mathcal{O}_K$, is said to be a Pólya field when the $\mathcal{O}_K$-algebra formed by the integer-valued polynomials on $\mathcal{O}_K$ admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when $K$ is not a Pólya field, we are interested in the embedding of $K$ in a Pólya field. We study here two notions which can be considered as measures of the obstruction for $K$ to be a Pólya field: the Pólya extensions $L/K$ where the characteristic ideals of $K$ extended to $L$ become principal, and the Pólya group which is the subgroup of the class group generated by the classes of the characteristic ideals.
LA - eng
KW - Pólya field; integral-valued polynomials; regular basis; Pólya extension
UR - http://eudml.org/doc/219724
ER -

References

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  12. G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern. J. Reine Angew. Math. 149 (1919), 97–116. Zbl47.0163.04
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