Counting Separable Polynomials in ℤ/n[x

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AbstractFor 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.