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Prem Prakash Pandey

Publications and source records attributed to Prem Prakash Pandey.

15 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

Rank of Pólya Groups in Lecacheux Parametric Family of Quintic Fields

In this article, we study the Pólya group of a new family of quintic fields, namely Lecacheux quintic fields. We show that the associated Pólya 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 Pólya 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 Pólya numbers of these fields in terms of the orders of their corresponding Pólya groups. In addition, we prove that several fields in this family are non-monogenic despite having index one.

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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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arXiv2023

Square-free values of polynomials

It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article, we propose a method to show that ``cyclotomic polynomials $Φ_{\ell}(X)$ take square free values with positive proportion". Following this method, conditionally, we do prove the Square-free conjecture for all cyclotomic polynomials. The method is applicable to some other class of polynomials as well. Moreover we show that the method readily succeeds for quadratic polynomials to give unconditional result. In the appendix, we give an elementary proof of the square-free conjecture for cyclotomic polynomials under abc conjecture.

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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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arXiv2017

On Freiman's 3k-4 theorem

One of the many theorems Freiman proved, in the second half of the twentieth century, in the subject which later came to be known as "structure theory of set addition", was 'Freiman's $3k-4$ theorem' for subsets of $\Z$. In this article we introduce concept of a new `structure' on finite subsets of integers. Sets with this structure are quite useful in additive number theory in some contexts. Also we give some criterion for subsets to posses this structure. Then this is used to establish an analog of Freiman's $3k-4$ theorem for the groups $\Z \times G,$ where $G$ is any abelian group.

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arXiv2017

3k-4 theorem for ordered groups

Recently, G. A. Freiman, M. Herzog, P. Longobardi, M. Maj proved two `structure theorems' for ordered groups \cite{FHLM}. We give elementary proof of these two theorems.

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arXiv2017

Distribution of prime ideals of higher residue degree across ideal classes in the class groups

In this article we investigate the distribution of prime ideals of residue degree bigger than one across the ideal classes in the class group of a number field $L$. A criterion for the class group of $L$ being generated by the classes of prime ideals of residue degree $f>1$ is provided. Further, some consequences of this study on the solvability of norm equations for $L/\mathbb{Q}$ and on the problem of finding annihilators for relative extensions are discussed.

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arXiv2013

On a Theorem of Deshouillers and Freiman

The study of `structure' on subsets of abelian groups, with small `doubling constant', has been well studied in the last fifty years, from the time Freiman initiated the subject. In \cite{DF} Deshouillers and Freiman establish a structure theorem for subsets of $\n$ with small doubling constant. In the current article we provide an alternate proof of one of the main theorem of \cite{DF}. Also our proof leads to slight improvement of the theorems in \cite{DF}.

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arXiv2012

Higher Residue Symbol

Given a prime number $l$ and a finite set of integers $S=\{a_1,...,a_m\}$ we find out the exact degree of the extension $\mathbb{Q}(a_1^{\frac{1}{l}},...,a_m^{\frac{1}{l}})/\mathbb{Q}$. We give an algorithm to compute this degree and then further relate it to the study of the distribution of primes $p$ for which all of $a_i$ assume a preassigned $l^{th}$ power residue simultaneously. Also we relate this degree to rank of a matrix obtained from $S=\{a_1,...,a_m\}$. This latter arguement enable one to describe the degree $\mathbb{Q}(a_1^{\frac{1}{l}},...,a_m^{\frac{1}{l}})/\mathbb{Q}$ in much simpler terms.

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arXiv2011

Higher residue symbols

Given a prime number $l$ and a finite set of integers $S=\{a_1,...,a_m\}$ we find out the exact degree of the extension $\mathbb{Q}(a_1^{\frac{1}{l}},...,a_m^{\frac{1}{l}})/\mathbb{Q}$. We give two different ways to compute this degree. The first method is using ramifiaction theory. The second proof follwos from our study of the distribution of primes $p$ for which all of $a_i$ are $l^{th}$ power residue simultaneously.

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