A necessary and sufficient condition for the existence of non-trivial $S_n$-invariants in the splitting algebra

TL;DR

This paper establishes a precise condition under which the invariants of the symmetric group acting on a polynomial's splitting algebra are trivial, linking algebraic properties of the base ring to group invariance.

Contribution

It proves that the vanishing intersection of annihilators of 2 and D_f in the base ring is both necessary and sufficient for trivial symmetric invariants in the splitting algebra.

Findings

Invariants under $S_n$ are trivial iff the intersection of annihilators of 2 and D_f is zero.

Provides a necessary and sufficient condition for the triviality of invariants.

Connects algebraic properties of the base ring with symmetry invariants in splitting algebras.

Abstract

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.