Pellian Forms and the Ideal Class Group: Unveiling Torsion and Rank
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2025
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| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | Pellian forms, generalizations of the classic Pell's equation, hold a profound connection to the algebraic structure of number fields, particularly their ideal class groups. This paper delves into the intricate relationship between Pellian forms and the ideal class group, with a specific focus on how solutions to these Diophantine equations can unveil the torsion and rank elements within the class group. The ideal class group, a finite abelian group, quantifies the extent to which unique factorization fails in a ring of integers, and its structure (its torsion subgroup and rank of its free abelian part) provides deep insights into the arithmetic of the underlying number field. We explore how the properties of fundamental units, derived from solutions to Pellian equations, directly influence the class number and the cyclic structure of the ideal class group. Through a comprehensive review of algebraic number theory, Diophantine equations, and class field theory, we establish a theoretical framework for understanding this connection. The methodology involves examining the group of units in real quadratic fields, their role in generating principal ideals, and how the existence and nature of solutions to Pellian forms can reveal specific elements of the ideal class group's 2-torsion or higher $p$-torsion. We also discuss computational techniques and theoretical implications for determining the full structure of the ideal class group. The findings suggest that Pellian forms offer a powerful, albeit complex, lens through which to investigate the often elusive properties of ideal class groups, providing novel avenues for understanding the arithmetic of number fields. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17804053 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Pellian Forms and the Ideal Class Group: Unveiling Torsion and Rank Revista, Zen MATH, 10 Pellian forms, generalizations of the classic Pell's equation, hold a profound connection to the algebraic structure of number fields, particularly their ideal class groups. This paper delves into the intricate relationship between Pellian forms and the ideal class group, with a specific focus on how solutions to these Diophantine equations can unveil the torsion and rank elements within the class group. The ideal class group, a finite abelian group, quantifies the extent to which unique factorization fails in a ring of integers, and its structure (its torsion subgroup and rank of its free abelian part) provides deep insights into the arithmetic of the underlying number field. We explore how the properties of fundamental units, derived from solutions to Pellian equations, directly influence the class number and the cyclic structure of the ideal class group. Through a comprehensive review of algebraic number theory, Diophantine equations, and class field theory, we establish a theoretical framework for understanding this connection. The methodology involves examining the group of units in real quadratic fields, their role in generating principal ideals, and how the existence and nature of solutions to Pellian forms can reveal specific elements of the ideal class group's 2-torsion or higher $p$-torsion. We also discuss computational techniques and theoretical implications for determining the full structure of the ideal class group. The findings suggest that Pellian forms offer a powerful, albeit complex, lens through which to investigate the often elusive properties of ideal class groups, providing novel avenues for understanding the arithmetic of number fields. |
| title | Pellian Forms and the Ideal Class Group: Unveiling Torsion and Rank |
| url | https://doi.org/10.5281/zenodo.17804053 |