arXiv subjects

Nimish Kumar Mahapatra

Publications and source records attributed to Nimish Kumar Mahapatra.

5 recordsLinked to original sources

arXiv2026

Rank of P\'olya Groups in Lecacheux Parametric Family of Quintic Fields

In this article, we study the P\'olya group of a new family of quintic fields, namely Lecacheux quintic fields. We show that the associated P\'olya groups can be arbitrarily large elementary abelian $5$-groups. Using density arguments, we prove that for every positive integer $k$, the set of odd integers $s$ such that the $5-$rank of the P\'olya group of the corresponding Lecacheux quintic field is at least $k$ has a positive density. Combining this with a result of Golod and Shafarevich, we see that for a positive proportion of $s$, the corresponding Lecacheux quintic fields admit an infinte $5-$class field tower. We also establish an upper bound for the P\'olya numbers of these fields in terms of the orders of their corresponding P\'olya groups. In addition, we prove that several fields in this family are non-monogenic despite having index one.

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arXiv2025

Discriminants and Large P\'olya Groups in Septic Number Fields

We investigate a new family of cyclic septic fields $\{K_t\}_{t\in\mathbb{Z}}$ arising from the Hashimoto--Hoshi construction. For this family, we compute the discriminant explicitly and characterize their P\'olya property under the condition that the polynomial $E(t) = t^{6} + 2t^{5} + 11t^{4} + t^{3} + 16t^{2} + 4t + 8$ takes fifth-power free values. We show that this family contains infinitely many non-P\'olya fields for which the cardinality of the P\'olya group is unbounded. We also establish that, assuming Bunyakovsky's conjecture for $E(t)$, this family contains infinitely many P\'olya fields. We further show that, for any fixed positive integer $m$, there exist infinitely many blocks of $m$ consecutive fields in this family whose cardinality of the P\'olya groups can be made arbitrarily large. Finally, we demonstrate that infinitely many fields in this family are non-monogenic with field index one.

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arXiv2025

Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture

There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ of $K$ is absolutely abelian. In this article, we explore the latter hypothesis: how often a number field $K$ has absolutely abelian Hilbert class field? For a number field $K$ to have absolutely abelian Hilbert class field, we obtain several criteria in terms of class number of $K$, P\'olya group of $K$, and genus number of $K$. We also show that for such number fields the $\ell-$torsion conjecture is true. Along with these, the article also reports some results on a theme to study class groups, where primes of higher degree are used to study class groups.

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arXiv2022

Non-Pólya Fields with Large Pólya Groups Arising from Lehmer Quintics

In this article we construct a new family of quintic non-Pólya fields with large Pólya groups. We study the upper bound of Pólya numbers of such fields and show that the Pólya numbers never exceed five times the size of its Pólya group. Finally we show that such non-Pólya fields are non-monogenic fields of field index one.

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arXiv2022

Primes of higher degree and Annihilators of Class groups

Let $L/K$ be a Galois extension of number fields with Galois group $G$. We discuss a new method to obtain elements in $\mathbb{Z}[G]$ which annihilate the class group of $L$. Using this method, we obtain annihilators of class groups of cyclotomic fields. We show that these annihilators are new. Some more consequences are also discussed. Moreover, we mention some results and connections to highlight importance of primes of higher residue degree.

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