A search for c-Wieferich primes

TL;DR

This paper investigates c-Wieferich primes over finite fields, providing an existence criterion and an algorithm for their computation, and explores their behavior in polynomial extensions.

Contribution

It introduces a new criterion and algorithm for identifying c-Wieferich primes in finite field polynomial rings and analyzes their properties in extensions.

Findings

Established an existence criterion for c-Wieferich primes.

Developed an algorithm for computing these primes.

Studied their behavior in finite field extensions.

Abstract

Let $q$ be a power of a prime number $p$, $\mathbb F_q$ be a finite field with $q$ elements and $\mathcal{G}$ be a subgroup of $(\mathbb F_q,+)$ of order $p$. We give an existence criterion and an algorithm for computing maximally $\mathcal{G}$-fixed c-Wieferich primes in $\mathbb F_q[T]$. Using the criterion, we study how c-Wieferich primes behave in $\mathbb F_q[T]$ extensions.