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Thread: quaternion algebra

  1. #1
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    quaternion algebra

    Let $\displaystyle \mathbb{H}$ be the quaternion algebra.

    Show that $\displaystyle \mathbb{H}$ 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?

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

    ????
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    MHF Contributor FernandoRevilla's Avatar
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    Re: quaternion algebra

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

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

    Similar reasoning if we suppose $\displaystyle y\neq 0$ .
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    Re: quaternion algebra

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

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

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

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

    Show that $\displaystyle \mathbb{H}$ 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?

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

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