  For a monic polynomial $f$ over a commutative, unitary ring $A$ the splitting
algebra $A_f$ is the universal $A$-algebra such that $f$ splits in $A_f$. The
symmetric group acts on the splitting algebra by permuting the roots of $f$. It
is known that if the intersection of the annihilators of the elements $2$ and
$D_f$ (where $D_f$ depends on $f$) in $A$ is zero, then the invariants under
the group action are exactly equal to $A$. We show that the converse holds.
