A necessary and sufficient condition for the existence of nontrivial Sn-invariants in the splitting algebra

Abstract

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