Number fields with large Pólya groups.
Saved in:
| Title: | Number fields with large Pólya groups. |
|---|---|
| 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] |
| Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
|---|---|
| Header | DbId: egs DbLabel: Engineering Source An: 194397959 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: Number fields with large Pólya groups. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Akbary%2C+Amir%22">Akbary, Amir</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> amir.akbary@uleth.ca</i><br /><searchLink fieldCode="AR" term="%22Maarefparvar%2C+Abbas%22">Maarefparvar, Abbas</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> abbas.maarefparvar@uleth.ca</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Nov2026, Vol. 288, p44-77. 34p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Quadratic+fields%22">Quadratic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Algebraic+number+theory%22">Algebraic number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Galois+theory%22">Galois theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: 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] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=194397959 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jnt.2026.03.004 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 34 StartPage: 44 Subjects: – SubjectFull: Quadratic fields Type: general – SubjectFull: Algebraic number theory Type: general – SubjectFull: Galois theory Type: general Titles: – TitleFull: Number fields with large Pólya groups. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Akbary, Amir – PersonEntity: Name: NameFull: Maarefparvar, Abbas IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 11 Text: Nov2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0022314X Numbering: – Type: volume Value: 288 Titles: – TitleFull: Journal of Number Theory Type: main |
| ResultId | 1 |
