arXiv subjects

Mahesh Kumar Ram

Publications and source records attributed to Mahesh Kumar Ram.

5 recordsLinked to original sources

arXiv2026

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 $K$, Pólya 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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arXiv2026

A primality test for $Kp^\ell - 1$ numbers

We develop an algebraic framework over arbitrary quadratic fields $L = \mathbb{Q}(\sqrt{D})$ to generalize the Miller-Rabin primality test. Consequently, we present a deterministic primality test for integers of the form $N = K p^{\ell} - 1$ that requires only a single modular exponentiation and achieves a computational complexity of $\tilde{\mathcal{O}}(\log^2 N)$. Furthermore, we also establish an analogue of Korselt's criterion within this setting. Finally, computational data generated using SageMath confirm its efficiency, successfully establishing the primality of numbers in the associated quadratic field within milliseconds.

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arXiv2025

Class numbers of Imaginary bicyclic biquadratic number fields

For any fixed positive integer $n$, we provide a method to compute all imaginary bicyclic biquadratic number fields with class number $n$, along with their class group structures, using the list of all imaginary quadratic number fields whose class numbers divide $2n$. We apply this method to list all imaginary bicyclic biquadratic number fields with class numbers $4$, $6$ and $7$. We also present the class group structure of each subfield of these fields.

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arXiv2023

Primes of Higher Degree

Let $K/\Q$ be a cyclic extension of number fields with Galois group $G$. We study the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree bigger than one in the class group of $K$. In particular, we explore such extensions $K/\Q$ for which there exist an integer $f>1$ such that the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree $f$ generate the full class group of $K$. It is shown that there are many such fields. These results are used to obtain information on class group of $K$; like rank of $\ell-$torsion of the class group, factors of class number, fields with class group of certain exponents, and even structure of class group in some cases. Moreover, such $f$ can be used to construct annihilators of the class groups.

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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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