2 results
Counting Separable Polynomials in ℤ/n[x]
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 2 / 01 June 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 346-352
- Print publication:
- 01 June 2018
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- Article
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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.
Bounds for the multiplicities of the roots of a complex polynomial
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 54 / Issue 3 / October 2011
- Published online by Cambridge University Press:
- 08 April 2011, pp. 587-598
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