Lehmer's totient problem over ${\mathbb{F}}_q[x]$

Abstract

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x],$ where ${\mathbb{F}}_q$ is a finite field with $q$ elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3,p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each of degree $1,2$ and $3$.

12 pages