# 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

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topLeriche, 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 -

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