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Math Help - Minimal Polynomials

  1. #1
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    Post Minimal Polynomials

    Hi

    I am wondering how to find the minimal polynomial of 2+\sqrt{i} over \mathbb{Q}.

    I have got it to x^{4}-2x^{2}+9 but I don't know how to shaw that this is irreducible!

    Thanks
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  2. #2
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    Quote Originally Posted by Kirsty View Post
    Hi

    I am wondering how to find the minimal polynomial of 2+\sqrt{i} over \mathbb{Q}.

    I have got it to x^{4}-2x^{2}+9 but I don't know how to shaw that this is irreducible!

    Thanks

    I don't know from where did you get that polynomial. I got:

    x=2+\sqrt{i}\Longrightarrow (x-2)^2=i\Longrightarrow (x-2)^4=-1\Longrightarrow x^4-8x^3+24x^2-32x+17 , and this polynomial is irreducible by Eisenstein's criterium with p=2 after carrying on the transformation x \rightarrow x+1

    Tonio
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  3. #3
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    Oops,

    I actually meant \sqrt{2}+i.

    Sorry
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