Units in Gauss integers ring

**tttcomrader** · November 1st 2008, 03:32 PM

Find units in a ring constituted by Gauss integers.

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

Now, of course $-1, 1, i, -i$ are units, but would $1-i,-1+i$ also be units? And are there anymore?

**ThePerfectHacker** · November 1st 2008, 03:33 PM

Originally Posted by tttcomrader

Find units in a ring constituted by Gauss integers.

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

Now, of course $-1, 1, i, -i$ are units, but would $1-i,-1+i$ also be units? And are there anymore?

No. A Gaussian is a unit if and only if $N(\alpha) = 1$ . Where $\alpha = a+bi$ and $N(\alpha) = a^2+b^2$ .

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