## Integral bases of the Gaussian integers

Let $E = \mathbb{Q}(i)$ and let $a,b,c,d \in \mathbb{Z}$. Prove that $B = \{a+bi, c+di\}$ is an integral basis for $E$ if and only if $ad-bc = \pm 1$.

Recall that the discriminant of $\mathbb{Q}(i)$ is $\mathsf{det} \left( \left[ \begin{array}{cc} 1 & i \\ 1 & -i \end{array} \right] \right)^2 = -4$, and that the conjugates of $a+bi$ are $a\pm bi$.

Note that the discriminant of $B$ is $\mathsf{det} \left( \left[ \begin{array}{cc} a+bi & c+di \\ a-bi & c-di \end{array} \right] \right)^2 = -4(ad-bc)^2$. By Theorem 6.10 in TAN, $B$ is an integral basis if and only if $-4(ad-bc)^2 = -4$, if and only if $ad-bc = \pm 1$.