Number Fields With Large Pólya Groups

Abstract

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