Algebraic elements in group rings

Abstract

In this brief note, we study algebraic elements in the complex group algebra C [ G ] {\mathbf {C}}[G] . Specifically, suppose ξ ∈ C [ G ] \xi \in {\mathbf {C}}[G] satisfies f ( ξ ) = 0 f(\xi ) = 0 for some nonzero polynomial f ( x ) ∈ C [ x ] f(x) \in {\mathbf {C}}[x] . Then we show that a certain fairly natural function of the coefficients of ξ \xi is bounded in terms of the complex roots of f ( x ) f(x) . For G G finite, this is a recent observation of [HLP]. Thus the main thrust here concerns infinite groups, where the inequality generalizes results of [K] and [W] on traces of idempotents.