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Irreducible polynomial

  1. Oct 31, 2006 #1
    Hi, I'm trying to show whether the polynomial

    [tex]g(x) = x^8+ x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 [/tex]

    is irreducible or not.

    So far I have evaluated [itex] g(x+1) [/itex] and applied Eisenstein's theorem to it. From what I gather it doesn't appear to be irreducible. Is this right, because I reckon it should be irreducible? This may just be a simple calculation error.

    And if g(x) is reducible, how do I go about reducing the polynomial more?

  2. jcsd
  3. Oct 31, 2006 #2


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    What is g(x)(x-1), and what are its roots?
  4. Oct 31, 2006 #3
    Of course, that would be helpful.

    [tex]g(x)(x-1) = (x^9-1)[/tex]

    and the roots are [itex] \alpha [/itex] existing in the complex numbers such that
    [itex] \alpha^9 = 1 [/itex]
    Last edited: Oct 31, 2006
  5. Oct 31, 2006 #4


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    So you can solve your problem now right?
  6. Nov 22, 2006 #5
    Two of the roots are cos(120)+isin(120)=w (cube root of 1,) and cos(240)+isin(240)=w^2. Combining these two roots (x-w)(x-w^2)=x^2+x+1.

    This then divides the polynominal giving: (1+x+x^2)(x^6+x^3+1).
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