On the invariants of the splitting algebra

Abstract

For a given monic polynomial $p(t)$ of degree $n$ over a commutative ring $k$, the splitting algebra is the universal $k$-algebra in which $p(t)$ has $n$ roots, or, more precisely, over which $p(t)$ factors, $p(t)=(t-\xi_1)...(t-\xi_n)$. The symmetric group $S_r$ for $1\le r\le n$ acts on the splitting algebra by permuting the first $r$ roots $\xi_1,...,\xi_r$. We give a natural, simple condition on the polynomial $p(t)$ that holds if and only if there are only trivial invariants under the actions. In particular, if the condition on $p(t)$ holds then the elements of $k$ are the only invariants under the action of $S_n$. We show that for any $n\ge 2$ there is a polynomial $p(t)$ of degree $n$ for which the splitting algebra contains a nontrivial element invariant under $S_n$. The examples violate an assertion by A. D. Barnard from 1974.