Wieferich Primes and Monogenic Trinomials

Abstract

A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1(\mathrm{mod}{p}^2)$. 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 $\{1,\hat{I}¸,\hat{I}{¸}^2,\dots ,\hat{I}{¸}^{N-1}\}$ is a basis for the ring of integers of $\mathbb{Q}(\hat{I}¸)$, where $f(\hat{I}¸)=0$. In this article, we show that ${\mathcal{F}}_p(x):=x^{2p}+2x^p+2$ is monogenic if and only if $p$ is not a Wieferich prime.