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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  1. P.J. Cahen, J.L. Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, vol 48, Amer. Math. Soc., Providence (1997). Zbl0884.13010MR1421321
  2. J.L Chabert, Factorial Groups and Pólya groups in Galoisian Extension of ℚ , Proceedings of the fourth international conference on commutative ring theory and applications (2002), 77-86. Zbl1107.13303MR2029819
  3. H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 2000. Zbl0786.11071MR1228206
  4. H. Furuya, Principal ideal theorem in the genus field for absolutely abelian extensions, J. Number Theory9 (1977), 4-15. Zbl0347.12006MR429820
  5. G. Gras, Class Field Theory, Springer, 2005. Zbl1019.11032MR1941965
  6. K. Hardy, R.H. Hudson, D. Richman, K.S. Williams and N.M. Holtz, Calculation of Class Numbers of Imaginary Cyclic Quartic Fields, Carleton-Ottawa Mathematical Lecture Note Series7 (1986), 201 pp. Zbl0615.12009MR906194
  7. D. Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1894-95), 175-546. 
  8. E.S. Golod, I.R. Shafarevich, On the class field tower, Izv. Akad. Nauk28 (1964), 261-272. Zbl0136.02602MR161852
  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
  13. K.S. Williams, Integers of biquadratic fields, Canad. Math. Bull.13 (1970), 519-526. Zbl0205.35401MR279069
  14. H. Zantema, Integer valued polynomials over a number field, Manusc. Math.40 (1982), 155-203. Zbl0505.12003MR683038

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