Consider the split monic $f=\prod\_{i=1}^n(x-x\_i)\in \mathbb Z[x\_1 ,\dots ,x\_n,x]$. Its discriminant is usually defined as $$(-1)^{n(n-1)/2}\prod\_{i=1}^nf^\prime(x\_i)=\prod\_{1\leq i<j\leq n}(x\_i-x\_j)^2.$$


What is the reason for taking this definition as opposed to $\prod\_{i=1}^nf^\prime(x\_i)$? The product of the derivatives at the roots "feels" to me more canonical than the product on the RHS.



Why the sign in the definition of the discriminant?
The reason is that the formula on the *right side* should be considered more fundamental, not the formula on the left, when seeking a symmetric expression in the roots. Don't use a product of anything "at" the roots, but a symmetric expression in the roots that vanishes if any pair of roots are equal. That explains the factors $(x\_i-x\_j)^2$. Do you consider the simplest polynomial with a double root at 0 to be $x^2$ or $-x^2$? The product on the left (without the sign) is also interesting and has a name: it is called the *resultant* of $f(x)$ and $f'(x)$.


