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Math Help - splitting over Z mod 2

  1. #1
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    splitting over Z mod 2

    let \alpha be a zero of f(x)=x^3+x^2+1 over \mathbb{Z}_{2}, Show that f(x) splits over \mathbb{Z}_{2}(\alpha)

    I don't understand this question because f(x) is irreducible in \mathbb{Z}_{2} ( f(0)=1 and f(1)=1) so how can \alpha be its zero.
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  2. #2
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    The zero does not lie in Z_2. It lies in an extension of Z_2 containing alpha.

    To help solve problem note that alpha^2 is also a root in Z_2(alpha).
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  3. #3
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    Quote Originally Posted by frankdent1 View Post
    let \alpha be a zero of f(x)=x^3+x^2+1 over \mathbb{Z}_{2}, Show that f(x) splits over \mathbb{Z}_{2}(\alpha)

    I don't understand this question because f(x) is irreducible in \mathbb{Z}_{2} ( f(0)=1 and f(1)=1) so how can \alpha be its zero.
    as whipflip15 suggested: show that if \alpha is a zero of x^3 + x^2 + 1 \in \mathbb{Z}_2[x], then: x^3+x^2+1=(x-\alpha)(x-\alpha^2)(x-\alpha^4).
    Last edited by NonCommAlg; April 13th 2009 at 10:48 PM.
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