Number fields with large Pólya groups.
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| Title: | Number fields with large Pólya groups. |
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| Authors: | Akbary, Amir1 (AUTHOR) amir.akbary@uleth.ca, Maarefparvar, Abbas1 (AUTHOR) abbas.maarefparvar@uleth.ca |
| Source: | Journal of Number Theory. Nov2026, Vol. 288, p44-77. 34p. |
| Subjects: | Quadratic fields, Algebraic number theory, Galois theory |
| Abstract: | The Pólya 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 Pólya 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 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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