The probability of relatively prime polynomials in $\mathbb{Z}_{p^k}[x]$

Abstract

Let $P_R\left(m,n\right)$ denote the probability that two randomly chosen monic polynomials $f$, $g\in R\left[x\right]$ of degrees $m$ and $n$, respectively, are relatively prime. Let $q=p^k$ be a prime power. We establish an explicit formula for $P_R\left(m,2\right)$ when $R={\mathbb{Z}}_q$, the ring of integers mod $q$.