Vol. 3, No. 2, 2010

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

Thomas R. Hagedorn and Jeffrey Hatley

Vol. 3 (2010), No. 2, 223–232
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 $R={ℤ}_{q}$, the ring of integers mod $q$.

Keywords
relatively prime polynomials
Mathematical Subject Classification 2000
Primary: 11C20, 13B25, 13F20