Discriminant of $x^4 - 7$

Question

According to the formula on the wikipedia page, the discriminant of $x^4 - 7$ should be 0. But the polynomial has roots $\pm \sqrt[4]{7}$ and $\pm i\sqrt[4]{7}$ so it is separable, so it shouldn't have zero discriminant. What's going on?

Answer 1

There is the term $256a^3e^3$, which is nonzero, and all other terms are zero.

Answer 2

Since $b=c=d=0$, the discriminant is: $$ \Delta=256 a^3 e^3 $$ In your case, you have $a=1$, $e=-7$, so: $$ \Delta = 256 \cdot (-7)^3 = -87808 \neq 0$$