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Math Help - Finite fields

  1. #1
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    Finite fields

    can any one help me to know how to do this question please:
    if I have theta is a root of the polynomial p(x) ,where p(x)=x^3-2x-2 is irreducible in Q[x] how can I compute (1+theta)(1+theta+theta^2) and (1+theta)/1+theta+theta^2 in Q(theta)
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  2. #2
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    Quote Originally Posted by Mike12 View Post
    can any one help me to know how to do this question please:
    if I have theta is a root of the polynomial p(x) ,where p(x)=x^3-2x-2 is irreducible in Q[x] how can I compute (1+theta)(1+theta+theta^2) and (1+theta)/1+theta+theta^2 in Q(theta)

    \displaystyle{p(\theta)=\theta^3-2\theta-2=0\Longrightarrow \theta^3=2\theta+2} , so for example

    \displaystye{(1+\theta)(1+\theta+\theta^2)=1+2\the  ta+2\theta^2+\theta^3=1+2\theta+2\theta^2+2\theta+  2=3+4\theta+2\theta^2}

    Tonio
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