Skew-Symmetric Elements of Rational Group Algebras

TL;DR

This paper provides a structure theorem for skew-symmetric elements in group algebras over algebraic extensions of the rationals, generalizing previous results in the area.

Contribution

It introduces a generalized structure theorem for skew-symmetric elements in group algebras over algebraic extensions of , extending prior work.

Findings

Established a structure theorem for skew-symmetric elements

Generalized known results to broader algebraic extensions

Enhanced understanding of involution-based element structures

Abstract

Let $RG$ be the group ring of a finite group $G$ over a commutative ring $R$ with $1$. An element $x$ in $RG$ is said to be skew-symmetric with respect to an involution $\sigma$ of $RG$ if $\sigma(x)=-x.$ A structure theorem for the skew-symmetric elements of $FG$ is given where $F$ is an algebraic extension of $\mathbb{Q}$ which generalizes some previously known results in this direction.