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Math Help - Irreducible polynomial and Isomorphism

  1. #1
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    Cool Irreducible polynomial and Isomorphism

    Hi there, I'm really stuck on this question, both parts i) and ii). As ii) rely's on i)

    Im able to show that x^2-2 is irreducible over Z[5] but am stuck in trying to show that it has a zero in F. I know that I need to use (r+sa)^2-2=0 and finding r and s but I can't factorize the polynomial adequately in Z[5]... any help would be great! And let me know if im on the wrong track?

    Thanks
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  2. #2
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    Re: Irreducible polynomial and Isomorphism

    Quote Originally Posted by bazzel View Post
    Hi there, I'm really stuck on this question, both parts i) and ii). As ii) rely's on i)

    Im able to show that x^2-2 is irreducible over Z[5] but am stuck in trying to show that it has a zero in F. I know that I need to use (r+sa)^2-2=0 and finding r and s but I can't factorize the polynomial adequately in Z[5]... any help would be great! And let me know if im on the wrong track?

    Thanks

    I think you need to check whether there exists

    \alpha=r+sa\in \mathbb{Z}_5(a)\,\,s.t.\,\,\alpha^2-2=0\iff (r+sa)^2-2=0\iff r^2+2rsa+s^2a^2-2=0\iff r^2+2rsa+3s^2-2=0.

    Now choose r=1\Longrightarrow 1+2sa+3s^2-2=0\iff 3s^2+2as-1=0\Longrightarrow s_{1,2}=-2a\pm \sqrt{-1}=-2a\pm 2

    For example one root is r+sa=1+(2-2a)a=1+2a-2a^2=2a
    Nemesis
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