Geodesic
Counting Separable Polynomials in Z/n[x]
Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 346-352

Voir la notice de l'article provenant de la source Cambridge University Press

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 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.
DOI : 10.4153/CMB-2017-013-4
Mots-clés : 13H05, 13B25, 13M10, separable algebra, separable polynomial 
Polak, Jason K. C. Counting Separable Polynomials in Z/n[x]. Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 346-352. doi: 10.4153/CMB-2017-013-4
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