On a class of Monogenic polynomials

3:00pm, Wednesday 23 July

Abstract

Let $f(x) \in \mathbb{Z}[x]$ be an irreducible polynomial of degree $n$ and $\theta$ be a root of $f(x)$. Let $K=\mathbb{Q}(\theta)$ be the number field and $\mathbb{Z}_K$ be the ring of algebraic integers of $K$. We say $f(x)$ is monogenic if $\{1, \theta, \ldots, \theta^{n-1} \}$ is a $\mathbb{Z}$-basis of $\mathbb{Z}_K$.

In this talk, we consider the family of polynomials $f(x)=x^{n-km} (x^k+a)^m+b \in \mathbb{Z}[x]$, $1\leq km<n$. We provide a necessary and sufficient conditions for $f(x)$ to be monogenic.  As an application, we get a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$, the symmetric group on $n$ symbols. This is a joint work with A. Jakhar and P. Yadav.