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Math Help - irreducible polynomials

  1. #1
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    irreducible polynomials

    Show that the polynomials  g = X^3 + X + 1 and  h = X^3 + X^2 + 1 are the only polynomials of degree 3 which are irreducible over F2.

    I know that they are both irreducible over F2 since they both have degree \leq 3 and have no roots in F2, but I'm stuck on how to show they're the only ones...can anyone please help?

    Thanks in advance!
    Last edited by hunkydory19; May 5th 2008 at 05:07 AM.
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  2. #2
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    Hello,

    Quote Originally Posted by hunkydory19 View Post
    Show that the polynomials  g = X^3 + X + 1 and  h = X^3 + X^2 + 1 are the only polynomials of degree 3 which are irreducible over F2.

    I know that they are both irreducible over F2 since they both have degree \leq 3 and have not roots in F2, but I'm stuck on how to show they're the only ones...can anyone please help?

    Thanks in advance!
    Because the other polynomials of degree 3 in F2 are x^3, x^3+1, x^3+x^2 and x^3+x and that they all have a root in F2
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    since they both have degree \leq 3
    Hey watch out. Degree 3 doesn't mean degree \leq 3. Polynomials have to contain the term X^3 here.
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