
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 is called a Wieferich prime if . A monic polynomial of degree is called monogenic if is irreducible over and is a basis for the ring of integers of , where . In this article, we show that is monogenic if and only if is not a Wieferich prime.
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