Related papers: On monogenity of certain number fields defined by …

For an algebraic number $\alpha$ of degree $n$, let $\mathcal{M}_{\alpha}$ be the $\mathbb{Z}$-module generated by $1,\alpha ,\ldots ,\alpha^{n-1}$; then $\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\,…

We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[\rho]]$ of a root $\rho$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a…

For pure extensions $K=\mathbb{Q}(\alpha)$ with $\alpha^n=m$, we give a short proof, based only on Dedekind's index theorem, of the $\alpha$-monogeneity criterion: $\mathbb{Z}[\alpha]=\mathcal{O}_K$ if and only if $m$ is square-free and…

We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F \in L[Y]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for…

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting…

We show that a monic univariate polynomial over a field of characteristic zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper leading coefficients by providing an explicit algorithm for computing the multiplicities…

We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $f\mod\mathfrak p$ does not converge for $\mathfrak p$ passing…

In 2012, for any integer $n \ge 2$, Kedlaya constructed an infinite class of monic irreducible polynomials of degree $n$ with integer coefficients having squarefree discriminants. Such polynomials are necessarily monogenic. Further, by…

A series of recent articles has shown that there exist only three monogenic cyclic quartic trinomials in ${\mathbb Z}[x]$, and they are all of the form $x^4+bx^2+d$. In this article, we conduct an analogous investigation for cubic…

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are…

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. We show that for all even $g…

For a field $K$, and a root $\alpha$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(\alpha)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

Let $q$ be a prime power. We estimate the number of tuples of degree bounded monic polynomials $(Q_1,\ldots,Q_v) \in (\mathbb{F}_q[z])^v$ that satisfy given pairwise coprimality conditions. We show how this generalises from monic…

We produce an explicit family of totally real cyclic quartic polynomials that are monogenic in many cases and, if the $abc$ conjecture holds, generate distinct monogenic quartic fields infinitely often. Additional families (also…

Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$…

Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde…