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.

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