Clarification about the definition for polynomial discriminant?

Question

On the wikipedia page about polynomial discriminants, it shows this definition: $$\Delta = a_n^{2n-2}\prod_{i<j}(r_i-r_j)^2$$ What I'm getting from this is that $\Delta$ is obviously the discriminant, $a$ is a coefficient, $n$ is the power of the polynomial (not sure on this one), and $r$ is a root of the polynomial. However I don't understand this business with $i$ and $j$. What's with the $i<j$ under the $\prod$? Why doesn't it have an upper bound?

Answer 1

The subscripts index the roots $r_i$ of the polynomial. Since we restrict to $i < j$ (which we could write as $1 \leq i < j \leq n$), this means that the product (1) runs only over pairs of distinct roots, and (2) counts each pair only once (as opposed to twice, for the two possible orders of two roots). For example, if $n = 3$, the pairs $(i, j)$ of integers such that $1 \leq i < j \leq n$ are $(1, 2)$, $(1, 3)$, and $(2, 3)$, and so the discriminant is $$\Delta = a_3^4 (r_1 - r_2)^2 (r_1 - r_3)^2 (r_2 - r_3)^2 .$$