Alternating-group polynomial invariants ($\mathbb{C}[X_1,\dots ,X_n{]}^{A_n}=\mathbb{C}[e_1,\dots ,e_n,\Delta ]$)

For the permutation action on $n$ coordinates with $n\ge 2$, every alternating-group invariant is a symmetric polynomial plus the Vandermonde product $\Delta$ times a symmetric polynomial. Thus the invariant ring is $\mathbb{C}[e_1,\dots ,e_n,Z]/(Z^2-D(e_1,\dots ,e_n))$, where $D$ is the discriminant polynomial. The two summands are independent over the symmetric polynomial ring.