= Alternating-group polynomial invariants
{title2=$\mathbb C[X_1,\ldots,X_n]^{A_n}=\mathbb C[e_1,\ldots,e_n,\Delta]$}
For the <permutation> action on $n$ coordinates with $n\geq2$, every alternating-group invariant is a <symmetric polynomial> plus the Vandermonde product $\Delta$ times a <symmetric polynomial>. Thus the invariant ring is $\mathbb C[e_1,\ldots,e_n,Z]/(Z^2-D(e_1,\ldots,e_n))$, where $D$ is the discriminant <polynomial>. The two summands are independent over the <symmetric polynomial> ring.
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