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arXiv · 2203.09342

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

Kevin Schlegel
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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.

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Kevin Schlegel. 2022-03-17. A necessary and sufficient condition for the existence of non-trivial $S_n$-invariants in the splitting algebra. https://arxiv.org/abs/2203.09342

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