Number Fields With Large P\'olya Groups

TL;DR

This paper investigates the structure of Pólya groups in various families of number fields, establishing finiteness and classification results for fields with specific Pólya indices, including explicit classifications under GRH.

Contribution

It provides new finiteness theorems and classifications for number fields with large Pólya groups, especially for imaginary bi-quadratic, tri-quadratic, and real quadratic fields.

Findings

Classified all imaginary bi-quadratic and tri-quadratic fields with Pólya index one.

Under GRH, listed all imaginary quadratic fields with Pólya index two.

Extended Dohmae's classification to extended R-D types.

Abstract

The P\'olya group ${\rm Po}(K)$ of a number field $K$ is the subgroup of the ideal class group ${\rm Cl}(K)$ of $K$ generated by the classes of all 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\'olya index $\left[{\rm Cl}(K):{\rm Po}(K)\right]$ in the families of Galois number fields, solvable 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\'olya index one. Furthermore, we classify all real quadratic fields of extended R-D type (with possibly only one more field) with the P\'olya index one. Also, under GRH, we give the complete list of 161 imaginary quadratic fields with the P\'olya 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.