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For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if$R[x]/f$ is a separable $R$-algebra. We derive formulae for the number ofseparable polynomials when $R = \mathbb{Z}/n$, extending a result of L.Carlitz. For instance, we show that the number of separable polynomials in$\mathbb{Z}/n[x]$ that are separable is $\phi(n)n^d\prod_i(1-p_i^{-d})$ where$n = \prod p_i^{k_i}$ is the prime factorisation of $n$ and $\phi$ is Euler'stotient function.

Counting Separable Polynomials in $\mathbb{Z}/n[x]$