Necessary and sufficient condition for reciprocal polynomials to be monogenic

Abstract

Let ${\mathbb{Z}}_K$ denote the ring of integers of the number field $K=\mathbb{Q}(\theta )$, where $\theta$ is a root of a monic irreducible polynomial $f(x)\in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if ${\mathbb{Z}}_K=\mathbb{Z}[\theta ]$. A polynomial $f(x)\in \mathbb{Z}[x]$ is called reciprocal if $f(x)=x^{°(f)}f(1/x)$. In this article, we establish necessary and sufficient conditions for the monogeneity of reciprocal polynomials. As an application, we obtain an alternative and much simpler proof that the maximal real subfields of cyclotomic fields are monogenic.