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


Published:20170413
Printed: Jun 2018
Jason K. C. Polak,
School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia
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 $\phi(n)n^d\prod_i(1p_i^{d})$
where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$
and $\phi$ is Euler's totient function.