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Counting separable polynomials in $\mathbb{Z}/n[x]$

  • Jason K. C. Polak,
    School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia
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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 $\phi(n)n^d\prod_i(1-p_i^{-d})$ 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 separable algebra, separable polynomial
MSC Classifications: 13H05, 13B25, 13M10 show english descriptions Regular local rings
Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
Polynomials
13H05 - Regular local rings
13B25 - Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
13M10 - Polynomials
 

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