> [!NOTE] Lemma (Unit Group of Ring of Gaussian Integers is Set of Fourth Roots of Unity) > The [[Unit Group of Ring|unit group]] of the [[Ring of Gaussian Integers|ring of Gaussian integers]] $\mathbb{Z}[i]$ is given by $\mathbb{Z}[i]^*=\{ 1,-1,i,-i \}.$ ###### Proof \[MA268\] Let $a,b\in \mathbb{Z}[i]$ such that $\alpha\beta=1$. Applying the fact that the [[Complex Modulus is Multiplicative|norm map is multiplicative]] yields $N(\alpha)N(\beta)=N(\alpha\beta)=N(1)=1$. Now $N(\alpha)$ and $N(\beta)$ are in $\mathbb{Z}$ and they multiply to give $1$. So either $N(\alpha)=N(\beta)=1$ or $N(\alpha)=N(\beta)=-1$. Write $\alpha=a+bi$ where $a,b\in \mathbb{Z}$. Then $a^2+b^2=\pm 1$. But $a,b$ are integers. So $(a,b)=(\pm1,0)$ or $(0,\pm1)$. Hence $\alpha=\pm 1$ or $\pm i$. Clearly $\pm 1$ and $\pm i$ are units. So the unit group is $\mathbb{Z}[i]^*=\{ 1,-1,i,-i \}.$