7. Given integers $a, b, c, d$. If the roots of the equation
$$
z^{4}+a z^{3}+b z^{2}+c z+d=0
$$

correspond to four points $A, B, C, D$ forming the vertices of a square in the complex plane, then the minimum value of the area of square $A B C D$ is $\qquad$
7. 2 .

Let the complex number corresponding to the center of the square be $m$. Then, after translating the origin of the complex plane to $m$, the vertices of the square are distributed on a circle, i.e., they are the solutions to the equation $(z-m)^{4}=n$ (where $n$ is a complex number).

From $z^{4}+a z^{3}+b z^{2}+c z+d=(z-m)^{4}-n$, comparing coefficients we know that $m=-\frac{a}{4}$ is a rational number.
Furthermore, from $-4 m^{3}=c$, we know that $m$ is an integer.
Then, from $d=m^{4}-n$, we know that $n$ is an integer.
Thus, the roots of the equation $(z-m)^{4}=n$ are
$$
z_{k}=m+\sqrt[4]{|n|}\left(\cos \frac{k \pi}{2}+\mathrm{i} \sin \frac{k \pi}{2}\right) .
$$

Therefore, the length of the diagonal of the square is $2 \sqrt[4]{|n|}$, and its area is
$$
2 \sqrt{|n|} \geqslant 2 \text {. }
$$