Mersenne banner
no. 2
Pólya fields and Pólya numbers
Amandine Leriche1
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, Volume 2 (2010) no. 2, pp. 21-26.

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 po K of a Pólya field containing K, namely the Pólya number of K.

Published online:
DOI: 10.5802/acirm.29
Classification: 11R04, 13F20, 11R16, 11R37
Keywords: Pólya fields, Hilbert class field, genus field, integer-valued polynomials
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 
@article{ACIRM_2010__2_2_21_0,
     author = {Amandine Leriche},
     title = {P\'olya fields and {P\'olya} numbers},
     journal = {Actes des rencontres du CIRM},
     pages = {21--26},
     publisher = {CIRM},
     volume = {2},
     number = {2},
     year = {2010},
     doi = {10.5802/acirm.29},
     zbl = {06938577},
     language = {en},
     url = {https://pmb.cedram.org/articles/10.5802/acirm.29/}
}
TY  - JOUR
AU  - Amandine Leriche
TI  - Pólya fields and Pólya numbers
JO  - Actes des rencontres du CIRM
PY  - 2010
SP  - 21
EP  - 26
VL  - 2
IS  - 2
PB  - CIRM
UR  - https://pmb.cedram.org/articles/10.5802/acirm.29/
DO  - 10.5802/acirm.29
LA  - en
ID  - ACIRM_2010__2_2_21_0
ER  - 
%0 Journal Article
%A Amandine Leriche
%T Pólya fields and Pólya numbers
%J Actes des rencontres du CIRM
%D 2010
%P 21-26
%V 2
%N 2
%I CIRM
%U https://pmb.cedram.org/articles/10.5802/acirm.29/
%R 10.5802/acirm.29
%G en
%F ACIRM_2010__2_2_21_0
Amandine Leriche. Pólya fields and Pólya numbers. Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 21-26. doi : 10.5802/acirm.29. https://pmb.cedram.org/articles/10.5802/acirm.29/

[1] P.J. Cahen, J.L. Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, vol 48, Amer. Math. Soc., Providence (1997). | Zbl

[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.

[3] H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 2000.

[4] H. Furuya, Principal ideal theorem in the genus field for absolutely abelian extensions, J. Number Theory 9 (1977), 4-15. | DOI | MR | Zbl

[5] G. Gras, Class Field Theory, Springer, 2005. | DOI | MR | Zbl

[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 Series 7 (1986), 201 pp. | Zbl

[7] D. Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1894-95), 175-546. | Zbl

[8] E.S. Golod, I.R. Shafarevich, On the class field tower, Izv. Akad. Nauk 28 (1964), 261-272. | Zbl

[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. | DOI | Zbl

[11] A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpren, J. Reine Angew. Math. 149 (1919), 117-124. | DOI | Zbl

[12] G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 97-116. | DOI | Zbl

[13] K.S. Williams, Integers of biquadratic fields, Canad. Math. Bull. 13 (1970), 519-526. | DOI | MR | Zbl

[14] H. Zantema, Integer valued polynomials over a number field, Manusc. Math. 40 (1982), 155-203. | DOI | MR | Zbl

Cited by Sources: