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Counting Separable Polynomials in ℤ/n[x]

Published online by Cambridge University Press:  20 November 2018

Jason K. C. Polak
Jason K. C. Polak*
Affiliation:
School of Mathematics and Statistics, _e University of Melbourne, Parkville, Victoria 3010, Australia, e-mail: [email protected]
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Abstract

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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.

Keywords

13H0513B2513M10separable algebraseparable polynomial
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 2 , 01 June 2018 , pp. 346 - 352
Copyright
Copyright © Canadian Mathematical Society 2018

References

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[2] Carlitz, L., The arithmetic of polynomiah in a Galoisfield. Amer. J. Math. 54(1932), no. 1, 39–50.http://dx.doi.Org/10.2307/2371075 Google Scholar
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