Function field analogues of Bang-Zsigmondy's theorem and Feit's theorem

TL;DR

This paper extends classical number theory theorems about prime divisors in exponential expressions to the setting of function fields, specifically within the Carlitz module, establishing analogous results and exceptions.

Contribution

It introduces Zsigmondy primes and large Zsigmondy primes in the Carlitz module context and proves their existence with explicit exceptions, extending classical theorems to function fields.

Findings

Established function field analogues of Bang-Zsigmondy's theorem.

Proved existence of large Zsigmondy primes in the Carlitz module.

Explicitly determined exceptional cases in the function field setting.

Abstract

In the number field context, Bang-Zsigmondy's theorem states that for any integers $u, m > 1$, there exists a prime divisor $p$ of $u^m - 1$ such that $p$ does not divide $u^n - 1$ for every integer $0 < n < m$ except in some exceptional cases that can be explicitly determined. A prime $p$ satisfying the conditions in Bang-Zsigmondy's theorem is called a Zsigmondy prime for $(u, m)$. In 1988, Feit introduced the notion of large Zsigmondy primes as follows: A Zsigmondy prime $p$ for $(u, m)$ is called a large Zsigmondy prime if either $p > m + 1$ or $p^2$ divides $u^m - 1$. In the same year, Feit proved a refinement of Bang-Zsigmondy's theorem which states that for any integers $u, m > 1$, there exists a large Zsigmondy prime for $(u, m)$ except in some exceptional cases that can be explicitly determined. In this paper, we introduce notions of Zsigmondy primes and large Zsigmondy primes in the Carlitz module context, and prove function field analogues of Bang-Zsigmondy's theorem and Feit's theorem.