On the Invariants of the Splitting Algebra

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 − ξ1) · · · (t − ξn). The symmetric group Sr for 1 ≤ r ≤ n acts on the splitting algebra by permuting the first r roots ξ1, . . . , ξr . 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 Sn. We show that for any n ≥ 2 there is a polynomial p(t) of degree n for which the splitting algebra contains a nontrivial element invariant under Sn. The examples violate an assertion by A. D. Barnard from 1974.