Related papers: On the monogenicity of power-compositional Shanks …

In this paper, for any nonic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^9+ax+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also…

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Ga\'al studied the…

Let $K = \Q(\th)$ be a number with $\th$ a root of an irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in \Z[x]$. In this paper, based on the $p$-adic Newton polygon techniques applied on decomposition of primes in number fields and…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ that is irreducible over ${\mathbb Q}$ is called cyclic if the Galois group over ${\mathbb Q}$ of $f(x)$ is the cyclic group of order $n$, while $f(x)$ is called monogenic if…

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…

Let $m$ be an integer, $m\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $\alpha$ be a root of \[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \] The totally real cyclic fields $K=Q(\alpha)$ are called simplest sextic fields and…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $f(x)\in\mathcal{O}_K[x]$ be a monic, irreducible polynomial. We establish necessary and sufficient conditions in terms of the critical points of $f(x)$ for the…

In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $\alpha$ of a monic irreducible polynomial $f(x)=x^8-m\in\mathbf Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper,…

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\mathbb{F}_{q}[x]$ the ring of polynomials over $\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\mathbb{F}_{q}[x]$ and $\Phi(k)$ the Euler function in…

In this paper we aim at constructing a sequence $\{\mathsf{M}_n^k(x)\}_{n\ge0}$ of $\mathbb R_{0,m}$-valued polynomials which are monogenic in $\mathbb R^{m+1}$ satisfying the Appell condition (i.e. the hypercomplex derivative of each…

In this paper, we deal with the problem of monogenity of number fields defined by monic irreducible trinomials $F(x)=x^{12}+ax^m+b\in \mathbb{Z}[x]$ with $1\leq m\leq11$. We give sufficient conditions on $a$, $b$, and $m$ so that the number…

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…

Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $\omega_k(f)$ denote the number of distinct monic irreducible factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the first and…

Let $\theta$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this…

Let $f(x) = \sum\limits _{i=0}^{n} a_i x^i $ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0 < a_0 \leq a_{1} \leq \cdots \leq a_{k-1} < a_{k} < a_{k+1} \leq \cdots \leq a_n$ for some $k$,…

Let $f(x) \in \mathbb{F}_p[x]$, and define the orbit of $x\in \mathbb{F}_p$ under the iteration of $f$ to be the set \[ \mathcal{O}(x):=\{x,f(x),(f\circ f)(x),(f\circ f\circ f)(x),\dots\}. \] An orbit is a $k$-cycle if it is periodic of…

Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(\theta)$ be an algebraic number field generated by a root $\theta$ of $f(x)$. In the present paper, for every rational prime $p$, our objective…

Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic…

For any quartic number field $K$ generated by a root $\alpha$ of an irreducible trinomial of type $x^4+ax^2+b\in Z[x]$, we characterize when $Z[\alpha]$ is integrally closed. Also for $p=2,3$, we explicitly give the highest power of $p$…