On the invariants of the splitting algebra

TL;DR

This paper investigates the invariants of the splitting algebra of a monic polynomial over a commutative ring, establishing conditions for trivial invariants and providing counterexamples to a previous assertion.

Contribution

It introduces a simple criterion for when only trivial invariants exist in the splitting algebra and constructs explicit examples with nontrivial invariants, challenging prior claims.

Findings

The invariant elements under the symmetric group action are trivial if and only if a specific condition on the polynomial holds.

There exist polynomials whose splitting algebra contains nontrivial invariants, contradicting earlier assertions.

The paper clarifies the structure of invariants in splitting algebras for monic polynomials over rings.

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.