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Thread: irreducible polynomial

  1. #1
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    irreducible polynomial

    Hi guys,

    I was wondering how I can prove that $\displaystyle x^4+x+2$ in the field $\displaystyle \mathbb{Z}_3[x]$ is irreducible?
    I know how to do it if the degree is 3 or less, but here I have no idea..

    Thanks in advance,
    Orbis
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Orbis View Post
    Hi guys,

    I was wondering how I can prove that $\displaystyle x^4+x+2$ in the field $\displaystyle \mathbb{Z}_3[x]$ is irreducible?
    I know how to do it if the degree is 3 or less, but here I have no idea..

    Thanks in advance,
    Orbis
    It's easy to see there are no linear factors.

    As for quadratic terms, WLOG its ok to assume the two constant terms are $\displaystyle 2 $ and $\displaystyle 1 $, since $\displaystyle -2=1 $ and $\displaystyle -1=2 $.

    So we have $\displaystyle x^4+x+2=(x^2+ax+2)(x^2+bx+1) = x^4+(a+b)x^3+abx^2+(a+b)x+2 $

    The $\displaystyle x^3 $ term tells us that $\displaystyle a+b\equiv0\bmod{3} $.

    But the $\displaystyle x $ term tells us that $\displaystyle a+b\equiv1\bmod{3} $.

    This is impossible, hence $\displaystyle x^4+x+2 $ is irreducible in $\displaystyle \mathbb{Z}/3\mathbb{Z} $.
    Last edited by chiph588@; Apr 4th 2010 at 06:49 PM.
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  3. #3
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    Ah, so simple! I was thinking along the lines of proving that $\displaystyle <x^4+x+2>$ was a maximal ideal, but that wasn't getting me anywhere.

    Thanks for the quick reply!

    Orbis
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