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arXiv · 1703.07064

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

Jason K. C. Polak
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Abstract

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 of separable 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 $ϕ(n)n^d\prod_i(1-p_i^{-d})$ where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$ and $ϕ$ is Euler's totient function.

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Jason K. C. Polak. 2017-03-21. Counting Separable Polynomials in $\mathbb{Z}/n[x]$. https://arxiv.org/abs/1703.07064

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