  Let $R=\mathbf{C}[\xi_1,\xi_2,\ldots]$ be the infinite variable polynomial
ring, equipped with the natural action of the infinite symmetric group
$\mathfrak{S}$. We classify the $\mathfrak{S}$-primes of $R$, determine the
containments among these ideals, and describe the equivariant spectrum of $R$.
We emphasize that $\mathfrak{S}$-prime ideals need not be radical, which is a
primary source of difficulty. Our results yield a classification of
$\mathfrak{S}$-ideals of $R$ up to copotency. Our work is motivated by the
interest and applications of $\mathfrak{S}$-ideals seen in recent years.
