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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?

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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 $$\Delta = a_3^4 (r_1 - r_2)^2 (r_1 - r_3)^2 (r_2 - r_3)^2 .$$

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  • $\begingroup$ What if $n=1$? How would that work for the product? $\endgroup$ – Sam Feb 21 '16 at 18:04
  • $\begingroup$ In that case there are no admissible pairs, so the product is empty, and usually we define the empty product to be $1$ (just like we usually define the empty sum to be $0$) and hence the discriminant to be $1$. This isn't so useful, however, since we already know that degree $1$ polynomials cannot have repeated roots. Note that for $n = 2$ this coincides with what we usually call the discriminant of a quadratic. $\endgroup$ – Travis Feb 21 '16 at 18:13

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