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Thread: Units in Gauss integers ring

  1. #1
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    Units in Gauss integers ring

    Find units in a ring constituted by Gauss integers.

    So I know that Gauss integers are $\displaystyle a+bi \ \ \ a,b \in \mathbb {Z} $[/tex]

    Now, of course $\displaystyle -1, 1, i, -i $ are units, but would $\displaystyle 1-i,-1+i$ also be units? And are there anymore?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Find units in a ring constituted by Gauss integers.

    So I know that Gauss integers are $\displaystyle a+bi \ \ \ a,b \in \mathbb {Z} $[/tex]

    Now, of course $\displaystyle -1, 1, i, -i $ are units, but would $\displaystyle 1-i,-1+i$ also be units? And are there anymore?
    No. A Gaussian is a unit if and only if $\displaystyle N(\alpha) = 1$. Where $\displaystyle \alpha = a+bi$ and $\displaystyle N(\alpha) = a^2+b^2$.
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