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Thread: Relatively Prime Quadratic Integers

  1. #1
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    Lightbulb Relatively Prime Quadratic Integers

    Hello Math Help Forum,

    I saw the following problem on another forum but the responses were all over the place and I was hoping if someone could give some clarity. The problem states:

    Assume $\displaystyle 32 = \alpha\beta$ for $\displaystyle \alpha,\beta$ relatively prime quadratic integers in $\displaystyle Q[i]$. It can be shown that $\displaystyle \alpha = \epsilon \gamma^2$ for some unit $\displaystyle \epsilon$ and some quadratic integer $\displaystyle \gamma$ in $\displaystyle Q[i]$.

    Can someone explain how this is shown?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well, $\displaystyle 32 = 2^5 = (1+i)^5(1-i)^5$. And $\displaystyle 1+i,1-i$ are irreducible. But $\displaystyle (1+i)=i(1-i)$. Hence $\displaystyle 32 = i(1-i)^{10}$ is the unique expression of $\displaystyle 32$ in $\displaystyle \mathbb{Z}[i]$ as a product of irreducibles. Can you finish from there?
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