arXiv subjects

Anuj Jakhar

Publications and source records attributed to Anuj Jakhar.

At least 19 recordsLinked to original sources

arXiv2026

A Deterministic Cryptographic Prime Generation Chain over Monogenic Cubic Number Fields and their Generalizations

Generating primes is a fundamental problem in modern cryptography. Deterministic primality tests work well for special integers such as Mersenne or Proth primes, but these forms are quite restrictive. In this paper, we give a direct method to construct new primes from known ones. Starting with a seed prime $q \equiv 1 \pmod{3}$, we construct an integer $N \equiv 1 \pmod{3}$ satisfying $(2N + 1)^2 \equiv -3 \pmod{q}$. We then prove that $N$ is prime using the structure of monogenic pure cubic fields $K = \mathbb{Q}(\sqrt[3]{d})$. The resulting test requires only a single modular exponentiation and runs in $\tilde{\mathcal{O}}(\log^2 N)$ time. Finally, we show how this construction extends to pure number fields of arbitrary prime degree.

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arXiv2026

Monogenic Fields from Polynomial Compositions with Applications

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[\alpha_i]]=1$, where $K_i=\mathbb{Q}(\alpha_i)$ and $\alpha_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.

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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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arXiv2026

On the distribution of shapes of totally real multiquadratic number fields

The shape of a number field $K$ of degree $m$ is defined as the equivalence class of the lattice of integers under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes $\mathscr{S}_{m-1} = \mathrm{GL}_{m-1}(\mathbb Z)\backslash \mathrm{GL}_{m-1}(\mathbb R)/\mathrm{GO}_{m-1}(\mathbb{R})$. The double quotient space is equipped with a natural measure $\mu$ which is induced from the Haar measure on $\mathrm{GL}_{m-1}(\mathbb R)$. We study the distribution of shapes of totally real multiquadratic number fields of degree $m:=2^n$ in which $2$ is unramified. We show that the distribution is governed by the restriction of $\mu$ to a certain torus orbit in $\mathscr{S}_{m-1}$. Our result resolves a conjecture of Haidar.

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arXiv2026

Arithmetic Aspects of Number Fields Generated by Polynomial Families

Let $f(x)=(x^{k}+c)^{m}-ax^{n}\in\mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$, where $k,m,n\in\mathbb{N}$ with $km>n$, and let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $f(x)$. We investigate the arithmetic properties of the number fields that arise from this family. We first obtain an explicit formula for the discriminant of $f(x)$. Using this formula, we establish necessary and sufficient conditions for the monogeneity of $f(x)$, expressed in terms of the prime divisors of $a$ and $c$ and the parameters $k,m,n$. This yields infinite families of monogenic polynomials of arbitrary degree, including families with a non-square-free discriminant. Building on these results, we extend our algebraic characterization to composite polynomials, establishing some explicit conditions for the monogeneity of the composition of $f(x)$ with an arbitrary polynomial $g(x)$. From an analytic point of view, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. We further study non-monogeneity via the field index $i(K)$ and, for each prime $p$, provide sufficient conditions ensuring $\nu_p(i(K))=1$, yielding partial progress toward a problem of Narkiewicz. We also highlight a connection with a class of differential equations naturally associated with $f(x)$. As an application, we determine the conditions under which the splitting field of $f(x)$ has a full symmetric Galois group. Several explicit examples illustrate our results.

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arXiv2026

On the distribution of shapes of octic Kummer extensions

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes $\mathscr{S}_{n-1} = \mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$. In this paper, we study the distribution of shapes of octic Kummer extensions $L=\mathbb{Q}(i,\sqrt[4]{m})$, where $m\in\mathbb{Z}[i]$ is fourth-power-free. We parametrize these shapes by explicit invariants known as shape parameters and establish an asymptotic formula for their joint distribution ordered by absolute discriminant. The limiting distribution is given by an explicit measure that factors as the product of a continuous measure and a discrete measure arising from local arithmetic conditions.

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arXiv2026

On the distribution of shapes of sextic pure number fields

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. The shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $\mathrm{GL}_{n-1}(\mathbb{Z}) \backslash \mathrm{GL}_{n-1}(\mathbb{R}) / \mathrm{GO}_{n-1}(\mathbb{R})$. We investigate the distribution of shapes of pure sextic number fields $K=\mathbb{Q}(\sqrt[6]{m})$, ordered by absolute discriminant. Such fields are partitioned into $20$ distinct Types determined by local conditions at $2$ and $3$, and an explicit integral basis is given in each case. For each Type, the shape of $K$ admits an explicit description in terms of shape parameters. Fixing the sign of $m$ and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure.

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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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arXiv2025

Behaviour of Newton Polygon over polynomial composition

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices of the Newton polygon of the composition $ g(f^n(x)) $ with respect to a prime $ p $ can be explicitly described in terms of the Newton polygon of the polynomial $ g(x) $. Our results provide deeper insights into how the Newton polygon of a polynomial evolves under iteration and composition, with applications to the study of dynamical irreducibility, eventual stability, non-monogenity of tower of number fields, etc.

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arXiv2024

A Study of monogenity of Binomial Composition

Let $θ$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, θ, θ^2, \ldots, θ^{n-1} \}$ is a basis for the ring $\Z_K$ of integers of $K = \Q(θ)$. In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup $\Z[θ]$ in $\Z_K$ where $K = \Q(θ)$ with $θ$ having minimal polynomial $F(x) = (x^m-b)^n - a \in \Z[x]$, $m\geq 1$ and $n \geq 2$. As an application, we provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the property that both $f(x)$ and $f(g(x))$ are monogenic.

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arXiv2023

A short note on number fields defined by exponential Taylor polynomials

Let $n$ be a positive integer and $f_n(x)= 1+x+\frac{x^2}{2!}+\cdots + \frac{x^n}{n!}$ denote the $n$-th Taylor polynomial of the exponential function. Let $K = \mathbf{Q}(θ)$ be an algebraic number field where $θ$ is a root of $f_n(x)$ and $\mathbf{Z}_K$ denote the ring of algebraic integers of $K$. In this paper, we prove that for any prime $p$, $p$ does not divide the index of the subgroup $\mathbf{Z}[θ]$ in $\mathbf{Z}_K$ if and only if $p^2\nmid n!$.

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arXiv2023

On the irreducibility of extended Laguerre Polynomials

Let $m\geq 1$ and $a_m$ be integers. Let $α$ be a rational number which is not a negative integer such that $α= \frac{u}{v}$ with $\gcd(u,v) = 1, v>0$. Let $ϕ(x)$ belonging to $\Z[x]$ be a monic polynomial which is irreducible modulo all the primes less than or equal to $vm+u$. Let $a_i(x)$ with $0\leq i\leq m-1$ belonging to $\Z[x]$ be polynomials having degree less than $\degϕ(x)$. Assume that the content of $(a_ma_0(x))$ is not divisible by any prime less than or equal to $vm+u$. In this paper, we prove that the polynomials $L_{m,α}^ϕ(x) = \frac{1}{m!}(a_mϕ(x)^m+\sum\limits_{j=0}^{m-1}b_ja_j(x)ϕ(x)^j)$ are irreducible over the rationals for all but finitely many $m$, where $b_j = \binom{m}{j}(m+α)(m-1+α)\cdots (j+1+α)~~~\mbox{ for }0\leq j\leq m-1$. Further, we show that $L_{m,α}^ϕ(x)$ is irreducible over rationals for each $α\in \{0, 1, 2, 3, 4\}$ unless $(m, α) \in \{ (1,0), (2,2), (4,4),(6,4)\}.$ For proving our results, we use the notion of $ϕ$-Newton polygon and some results from analytic number theory. We illustrate our results through examples.

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arXiv2023

On Schur's irreducibility results and generalised $ϕ$-Hermite polynomials

Let $c$ be a fixed integer such that $c \in \{0,2\}.$ Let $n$ be a positive integer such that either $n\geq 2$ or $2n+1 \neq 3^u$ for any integer $u\geq 2$ according as $c = 0$ or not. Let $ϕ(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than $2n+c$. Let $a_i(x)$ with $0\leq i\leq n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\degϕ(x)$. Let $a_n \in \mathbb{Z}$ and the content of $(a_na_0(x))$ is not divisible by any prime less than $2n+c$. For a positive integer $j$, if $u_j$ denotes the product of the odd numbers $\leq j$, then we show that the polynomial $\frac{a_{n}}{u_{2n+c}}ϕ(x)^{2n}+\sum\limits_{j=0}^{n-1}a_j(x)\frac{ϕ(x)^{2j}}{u_{2j+c}}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{j=0}^{n}a_j\frac{x^{2j}}{u_{2j+c}}$ with $a_j \in \mathbb{Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.

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arXiv2023

An extension of a second irreducibility theorem of I. Schur

Let $n \neq 8$ be a positive integer such that $n+1 \neq 2^u$ for any integer $u\geq 2$. Let $ϕ(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n+1$. Let $a_j(x)$ with $0\leq j\leq n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\degϕ(x)$. Assume that the content of $(a_na_0(x))$ is not divisible by any prime less than or equal to $n+1$. In this paper, we prove that the polynomial $f(x) = a_n\frac{ϕ(x)^n}{(n+1)!}+ \sum\limits_{j=0}^{n-1}a_j(x)\frac{ϕ(x)^{j}}{(j+1)!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{j=0}^{n}a_j\frac{x^{j}}{(j+1)!}$ with $a_j \in \mathbb{Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.

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arXiv2023

DISCRIMINANT and integral basis OF $\mathbb{Q}(\sqrt[12]{a})$

Suppose $m$ be a $12$-th power free integer. Let $K=\mathbb{Q}(θ)$ be an algebraic number field defined by a complex root $θ$ of an irreducible polynomial $x^{12}-m$ and $O_K$ be its ring of integers. In this paper, we determine the highest power of $p$ dividing the index of the subgroup $\Z[θ]$ in $O_K$ and $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to the construction of an integral basis of $K$ which is illustrated with examples. In particular, when $m$ is a square free integer, we provide necessary and sufficient conditions for the set $\{1,θ,θ^2,\cdots,θ^{10},θ^{11}\}$ to be an integral basis of $K.$

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arXiv2023

On power basis of a class of number fields

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(θ)$ be an algebraic number field defined by a complex root $θ$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The aim of this paper is to provide the necessary and sufficient conditions involving only $a,c$ and $n$ for a given prime $p$ to divide the index of the subgroup $\Z[θ]$ in $\Z_K$. As a consequence, we provide families of monogenic algebraic number fields. Further, when $\Z_K \neq \Z[θ]$, we determine explicitly the index $[\Z_K : \Z[θ]]$ in some cases.

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arXiv2023

ON the index divisors of certain number fields

Let $K=\Q(θ)$ be an algebraic number field with $θ$ a root of an irreducible quadrinomial $f(x) = x^6+ax^m+bx+c\in\Z[x] $ with $m\in\{2,3,4,5\}$. In the present paper, we give some explicit conditions involving only $a,~b,~c$ and $m$ for which $K$ is non-monogenic. In each case, we provide the highest power of a rational prime $p$ dividing index of the field $K$. In particular, we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for these number fields. Finally, we illustrate our results through examples.

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arXiv2023

On common index divisor of the number fields defined by $x^7+ax+b$

Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(θ)$ be an algebraic number field generated by a root $θ$ of $f(x)$. In the present paper, for every rational prime $p$, our objective is to determine the necessary and sufficient conditions involving only $a,~b$ so that $p$ is a divisor of the index of the field $K$. In particular, we provide sufficient conditions on $a$ and $b$, for which $K$ is non-monogenic. In a special case, we show that if either $8$ divides both $a\pm1$, $b$ or $32$ divides both $a+4$, $b$, then $K$ is non-monogenic. We illustrate our results through examples.

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