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Counting Separable Polynomials in ℤ/n[x]

Published online by Cambridge University Press:  20 November 2018

Jason K. C. Polak*
Affiliation:
School of Mathematics and Statistics, _e University of Melbourne, Parkville, Victoria 3010, Australia, e-mail: jpolak@jpolak.org

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 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

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References

[1] Auslander, M. and Goldman, O., The Brauer group ofa commutative ring. Trans. Amer. Math. Soc. 97(1960), no. 3, 367409.http://dx.doi.org/10.1090/S0002-9947-1960-0121392-6 CrossRefGoogle Scholar
[2] Carlitz, L., The arithmetic of polynomiah in a Galoisfield. Amer. J. Math. 54(1932), no. 1, 3950.http://dx.doi.Org/10.2307/2371075 CrossRefGoogle Scholar
[3] DeMeyer, F. and Ingraham, E., Separable algebras over commutative rings. Lecture Notes in Mathematics, 181, Springer-Verlag, Berlin-New York, 1971.Google Scholar
[4] Magid, A. R., The separable Galois theory of commutative rings. Second ed., Pure and Applied Mathematics, CRC Press, Boca Raton, FL, 2014.http://dx.doi.Org/10.1201/b17145 Google Scholar
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