.png){kind=icon}
.png){kind=icon}
Return
Number fields with large Pólya groups ☆
.png){kind=icon}
PRE
AI
DOI：10.1016/j.jnt.2026.03.004.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
The P & oacute;lya group Po(K) of a number field K is the subgroup of the ideal class group Cl(K) of K generated by the classes of the products of the prime ideals of K with the same norm. Motivated by the classical "one class in each genus problem", we prove general finiteness theorems for the number fields K with a fixed P & oacute;lya index [Cl(K) : Po(K)] in the families of Galois number fields, CM-fields, and real quadratic fields of extended R-D type. We also give classification results for specific families. Most notably, we classify, unconditionally, all imaginary bi-quadratic and imaginary tri-quadratic fields with the P & oacute;lya index one. Furthermore, we classify all real quadratic fields of extended R-D type (with possibly only one more field) with the P & oacute;lya index one. Also, under GRH, we give the complete list of 161 imaginary quadratic fields with the P & oacute;lya index two. Finally, as a byproduct of our results, we extend, from narrow R-D types to the extended R-D types, Dohmae's classification of real quadratic fields of narrow R-D type whose narrow genus numbers equal their narrow class numbers. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
Keywords:
P & oacute
lya groups
Genus fields
One class in each genus
Quadratic fields of R-D types
Journal
J
IF:
0.7
Papers:
103
Citations:
3.3K