Is the sign of the discriminant of a polynomial important? The discriminant of a polynomial is a very useful quantity but I have never seen the sign of a discriminant be useful. I know that the generalization of a discriminant from number fields over $\Bbb Q$ is other number fields is an ideal defined over the base field and clearly, in this generalized case, the sign cannot matter.
However, perhaps the sign is important elsewhere?
 A: The sign of the discriminant of a monic polynomial $f(X) \in \mathbb R[X]$ is
$$(-1)^{r}$$
where $r$ is half the number of non-real roots of $f$. 
Consider for instance a cubic monic polynomial $f(X)$ with roots $a, b, c$. Since non-real roots appear in complex conjugate pairs there are two possibilities: either all of $a, b, c$ are real, or one is real and the other two are complex conjugates. 
The discriminant is
$$(a-b)^2(a-c)^2(b-c)^2.$$
This is clearly positive if all of $a,b,c$ are real. In the case where $a$ and $b$ are complex conjugate and $c$ is real, then $a-c$ and $b-c$ are complex conjugate so $(a-c)^2(b-c)^2$ is a positive real number, and on the other hand $(a-b)^2$ is the square of a purely imaginary number and is therefore negative, so the discriminant is negative. 
(By the way, this is cool because you can calculate the discriminant of $f$ from its coefficients, and thus you have an easy and direct way to check whether a cubic polynomial has $3$ real roots or only $1$ real root.)
