1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Multinacci Polynomial
  1. Feral Polynomials (Replies: 3)