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Counting Separable Polynomials in ℤ/n[x]

  • Jason K. C. Polak (a1)
Abstract

For a commutative ring R, a polynomialf ∈ 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 = /n, extending a result of L. Carlitz. For instance, we show that the number of polynomials in /n[x] that are separable is ϕ(n)nd Πi(1 − ), where n = is the prime factorisation of n and ϕ is Euler’s totient function.

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[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
[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
[3] DeMeyer, F. and Ingraham, E., Separable algebras over commutative rings. Lecture Notes in Mathematics, 181, Springer-Verlag, Berlin-New York, 1971.
[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
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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