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Polynomials in Zn

  1. Oct 27, 2011 #1
    I am taking a first course in algebra and this is a problem in my textbook that has me stumped:

    [tex]Fix \ a \in \mathbb{Z}_{n} \ and \ f \in \mathbb{Z}_{n}[x] \ with \ \deg(f) = m. Show \ there \ is \ h_{m-1} \in \mathbb{Z}_{p}[x] \ with \ \deg(h_{m-1}) \leq (m-1) \ so \ f = a_{m}(x - a)^m + h_{m-1}[/tex]

    We haven't covered polynomial rings or anything like that.

    To be honest I don't understand how it makes sense to add coefficients in Zn with coefficients in Zp, although maybe it makes sense if p | n?

    If it's true then:
    [tex] f(a) = \sum_{k = 0}^{m} a_{k}a^{k} = h_{m-1}(a) [/tex]
    But it's unclear to me how that's possible.

    Any hint at all is really appreciated.
  2. jcsd
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