Email this article Pólya groups and fields in some real biquadratic number fields
- Authors: El Madrari S.1
-
Affiliations:
- Faculty of Sciences and Techniques Moulay Ismail University
- Issue: Vol 30, No 150 (2025)
- Pages: 128-143
- Section: Original articles
- URL: https://bakhtiniada.ru/2686-9667/article/view/298082
- ID: 298082
Cite item
Full Text
Abstract
Let K be a number field and \( O_K \) be its ring of integers. Let \( Π_q (K) \) be the product of all prime ideals of \( O_K \) with absolute norm q. The Pólya group of a number field is the subgroup of the class group of K generated by the classes of \( Π_q (K) \). K is a Pólya field if and only if the ideals \( Π_q (K) \) are principal. In this paper, we follow the work that we have done in [S. EL Madrari, “On the Pólya fields of some real biquadratic fields”, Matematicki Vesnik, online 05.09.2024] where we studied the Pólya groups and fields in a particulare cases. Here, we will give the Pólya groups of \( K=Q(√(d_1 ),√(d_2 )) \) such that \( d_1=lm_1 \) and \( d_2=lm_2 \) are square-free integers with \( l>1 \) and \( gcd(m_1;m_2)=1 \) and the prime 2 is not totally ramified in \( K⁄Q \). And then, we characterize the Pólya fields of the real biquadratic fields K.
About the authors
Said El Madrari
Faculty of Sciences and Techniques Moulay Ismail University
Author for correspondence.
Email: saidelmadrari@gmail.com
ORCID iD: 0000-0003-1632-8441
PhD
Morocco, BP 509 Boutalamine, Errachidia, MoroccoReferences
- G. Pólya, "Über ganzwertige Polynome in algebraischen Zahlkörpren", Journal Für die Reine und Angewandte Mathematik, 149 (1919), 97-116.
- A. Ostrowski, "Über ganzwertige Polynome in algebraischen Zahlkörpren", Journal Für die Reine und Angewandte Mathematik, 149 (1919), 117-124.
- H. Zantema, "Integer valued polynomials over a number field", Manuscripta Mathematica, 40:1-2 (1982), 155-203.
- P.J. Cahen, J.L. Chabert, Integer-Valued Polynomials, Mathematical Surveys and Monographs, 48, Amer. Math. Soc., Providence, 1997.
- A. Leriche, "Pólya fields, pólya groups and pólya extensions: A question of capitulation", Journal de Théorie des Nombres de Bordeaux, 23 (2011), 235-249.
- B. Heidaryan, A. Rajaei, "Some non-Pólya biquadratic fields with low ramification", Revista Matematica Iberoamericana, 33:3 (2017), 1037-1044.
- Ch. Wend-Waoga Tougma, "Some questions on biquadratic Pólya fields", Journal of Number Theory, 229 (2021), 386-398.
- A. Maarefparvar, "Pólya group in some real biquadratic fields", Journal of Number Theory, 228 (2021), 1-7.
- S. El Madrari, "On the Pólya fields of some real biquadratic fields", Matematicki Vesnik, http://www.vesnik.math.rs/inpress/mv2023 069.pdf.
- C. Bennett Setzer, "Units over totally real C_2×C_2 fields", Journal of Number Theory, 12:2 (1980), 160-175.
- T. Kubota, "Über den bizyklischen biquadratischen Zahlkörper", Nagoya Mathematical Journal, 10 (1956), 65-85.
- K.S. Williams, "Integers of biquadratic fields", Canadian Mathematical Bulletin, 13:4 (1970), 519-526.
- E. Haught, Bicyclic Biquadratic Number Fields, Masters Thesis, VPI&SU, Blacksburg, 1972.
Supplementary files
