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Math Help - Irreducible polynomium

  1. #1
    Junior Member
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    Irreducible polynomium

    Hi everyone, I got stuck in this exercise, where I am to show that the polynomial g(x)=x^3+x+1 is irreducible over \mathbb{Q}[x].

    In this case you can't use Eisenstein, so what was it that the alternative looked like?

    I tried to use polynomial division, assuming it had a root and then tried to reach a contradiction, but that just got me where I started:
    x^3+x+1=(x-a)(x^2+ax+1+a^2)+1+a+a^3 and then you still had to argue, that 1+a+a^3 couldn't be a nonzero, rational number.
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  2. #2
    Senior Member
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    Hi

    A polynomial in \mathbb{Q}[X] whose degree is 2 or 3 is reducible if and only if it has a root. Moreover, roots in \mathbb{Q} of a monic polynomial P\in\mathbb{Q}[X] are integers which divide P(0).

    Therefore in your case, if X^3+X+1 has a root in \mathbb{Q}, it's 1 or -1. Since they're not roots, X^3+X+1 has no root and is irreducible over \mathbb{Q}[X].
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  3. #3
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    Ahh yes, I can see that.
    Thanks a lot.
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