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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.

We refine a result of Dubickas on the maximal multiplicity of the roots of a complex polynomial, and obtain several separability criteria for complex polynomials with large leading coefficient. We also give p-adic analogous results for polynomials with integer coefficients.

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Counting Separable Polynomials in ℤ/n[x]

Bounds for the multiplicities of the roots of a complex polynomial

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2 results

Counting Separable Polynomials in ℤ/n[x]

Bounds for the multiplicities of the roots of a complex polynomial