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

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.