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

Published:2017-04-13
Printed: Jun 2018
• Jason K. C. Polak,
School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia
 Format: LaTeX MathJax PDF

## 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 $\phi(n)n^d\prod_i(1-p_i^{-d})$ where $n = \prod p_i^{k_i}$ is the prime factorisation of $n$ and $\phi$ is Euler's totient function.
 Keywords: separable algebra, separable polynomial
 MSC Classifications: 13H05 - Regular local rings 13B25 - Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13M10 - Polynomials

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