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Math Help - quaternion algebra

  1. #1
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    quaternion algebra

    Let \mathbb{H} be the quaternion algebra.

    Show that \mathbb{H} is a division algebra, i.e.

    if xy = 0, then x = 0 or y = 0.

    I understand this but I am not sure how to prove it. Is it as simple as just multiplying by x and y inverse?

    x^{-1}xy=y=0 and xyy^{-1}=x=0

    ????
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: quaternion algebra

    Essentially you are right. Suppose x,y\in\mathbb{H} and xy=0 . We have to prove x=0 or y=0 . If x\neq 0 then

    xy=0\Rightarrow x^{-1}(xy)=0\Rightarrow (x^{-1}x)y=0\Rightarrow 1y=0\Rightarrow y=0

    Similar reasoning if we suppose y\neq 0 .
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  3. #3
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    Re: quaternion algebra

    q\in\mathbb{H}, \ q\neq 0

    How do I go about showing that \{q, \ qi, \ qj, \ qk\} is a basis of \mathbb{H}\mbox{?}
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: quaternion algebra

    \lambda_1q+\lambda_2(qi)+\lambda_3(qj)+\lambda_4(q  k)=0 implies q(\lambda_11+\lambda_2i+\lambda_3j+\lambda_4k)=0 . Now use that \mathbb{H} is a division algebra.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Re: quaternion algebra

    Quote Originally Posted by dwsmith View Post
    Let \mathbb{H} be the quaternion algebra.

    Show that \mathbb{H} is a division algebra, i.e.

    if xy = 0, then x = 0 or y = 0.

    I understand this but I am not sure how to prove it. Is it as simple as just multiplying by x and y inverse?

    x^{-1}xy=y=0 and xyy^{-1}=x=0

    ????
    Technically you're asked to prove that \mathbb{H} is an integral domain. You could just note that there exists a multipicative norm on \mathbb{H} which satisfies |x|=0 if and only if x=0.
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