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Multinacci Polynomial

  1. Nov 3, 2006 #1
    Can anyone please give me a hint on how I can prove that

    [tex] g(x) = x^m - x^{m-1} - x^{m-2} - ... - x - 1 [/tex]

    is irreudicble over the rationals for all natural numbers m?

    Last edited: Nov 3, 2006
  2. jcsd
  3. Nov 3, 2006 #2
    I would use an evaluation homomorphism, to change the variable, and then use the Eisenstein criterion for a prime p and show it is irreducible. I haven't actually done it (so this might be wrong), but this is what I would try first.

    The evaluation homomorphism I would use would be of the form

    [tex]\phi_{x+1}\,:\,\mathbb{Q}[x] \rightarrow \mathbb{Q}[x] \quad\quad \phi_{x+1}(g(x)) = g(x+1)[/tex]

    Hopefully you'll get a nice binomial expression to which you can check if p divides (by Eisenstein's Criteria).
  4. Nov 8, 2006 #3
    Alas, the Eisenstein criteria will not work here for every m.

    Indeed [tex] \phi_{x+1}(x^4 - x^3 - x^2 - x - 1) = x^4 + 3x^3 + 2x^2 - 2x - 3 [/tex]

    and then (2,3) = 1 implies that no suitable prime can be found.

    Any other ideas?
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