Wieferich Primes and Monogenic Trinomials

TL;DR

This paper establishes a connection between Wieferich primes and the monogenicity of certain trinomials, showing that the polynomial ${ m{F}}_p(x)$ is monogenic if and only if $p$ is not a Wieferich prime.

Contribution

It proves a precise criterion linking Wieferich primes to the monogenicity of a specific family of trinomials, expanding understanding of algebraic number theory.

Findings

${ m{F}}_p(x)$ is monogenic iff $p$ is not a Wieferich prime.

Provides a new characterization of Wieferich primes through polynomial monogenicity.

Establishes a direct link between prime properties and algebraic number field bases.

Abstract

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 $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. 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.