Abstract
Groups of order 4 are isomorphic to either \({\mathbb {Z}}/4{\mathbb {Z}}\) or \({\mathbb {Z}}/2{\mathbb {Z}} \times {\mathbb {Z}}/2{\mathbb {Z}}\). We give certain sufficient conditions permitting to specify the structure of class groups of order 4 in the family of real quadratic fields \({\mathbb {Q}}{(\sqrt{n^2+1})}\) as n varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point \(-1\). As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of n.
Résumé
À isomorphisme près, il y a deux groupes possibles d’ordre 4: \({\mathbb {Z}}/4{\mathbb {Z}}\) et \({\mathbb {Z}}/2{\mathbb {Z}}\times \mathbb {Z}/2{\mathbb {Z}}\). Nous donnons des conditions suffisantes permettant de spécifier la structure des groupes de classes d’ordre 4 dans la famille des corps quadratiques réels \({\mathbb {Q}}{(\sqrt{n^2+1})}\) lorsque n parcourt l’ensemble des entiers positifs. De plus, nous calculons la valeur de la fonction zêta de Dedekind attachée à ces corps au point \( -1 \). Comme résultat secondaire, nous montrons que la cardinalité du groupe de classes des corps de cette famille peut être aussi grande que possible en augmentant le nombre de facteurs premiers impairs distincts de n.
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Acknowledgements
The authors would like to express their gratitude to Professor Claude Levesque for carefully reading this manuscript and for his useful comments. The second author is grateful to Professor Srinivas Kotyada for stimulating environment at The Institute of Mathematical Sciences, Chennai during his visiting period. The authors are thankful to the anonymous referees for their valuable comments and suggestions which have helped improving the presentation immensely. The second author acknowledges the grant SERB MATRICS Project (No. MTR/2017/00100). The third author is partially supported by ‘Infosys grant’.
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Chakraborty, K., Hoque, A. & Mishra, M. On the structure of order 4 class groups of \({\mathbb {Q}}(\sqrt{n^2+1})\). Ann. Math. Québec 45, 203–212 (2021). https://doi.org/10.1007/s40316-020-00139-1
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DOI: https://doi.org/10.1007/s40316-020-00139-1