Related papers: On Monogeneity of reciprocal polynomials

The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $\theta$ is the index $[\mathbb{Z}_K:\mathbb{Z}[\theta]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(\theta)$.…

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of…

Let $f(x) = (x^{2}+1)^{n} - a x^{n} \in \mathbb{Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta$ be a root of $f(x)$, set $K= \mathbb{Q}(\theta)$, and denote by $\mathbb{Z}_{K}$ the ring of integers of $K$. The index of $f$, denoted…

Let $\theta$ 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 $\{ 1, \theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring $\Z_K$ of integers…

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 $f(x)\in\mathbb{Z}[x]$…

Let $f(x)\in {\mathbb Z}[x]$ be an $N$th degree polynomial that is monic and irreducible over ${\mathbb Q}$. We say that $f(x)$ is {\em monogenic} if $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of…

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb…

Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$. We say $f(x)$ is \emph{monogenic} if $\Theta=\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers ${\mathbb…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

We call a polynomial monogenic if a root $\theta$ has the property that $\mathbb{Z}[\theta]$ is the full ring of integers in $\mathbb{Q}(\theta)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any…

Suppose that $f(x)\in {\mathbb Z}[x]$ is monic and irreducible over ${\mathbb Q}$ of degree $N$. We say that $f(x)$ is monogenic if $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$,…

A polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called \emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

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 $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where…

A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod{p^2}$. A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and…