Characters of Finite Groups, Algebraic Integers, and Burnside

The goal of this note is to write down the proofs of results about characters of irreducible representations of finite groups, and specifically, the result (due to Burnside) that each of them vanishes at at least one element of the group. We present various results told to us by various people, and conclude with two theorems by Gallagher, which strengthen or generalize Burnside’s theorem for finite groups. As an aside, we write down several other facts (from Isaacs’s book [Isa]) involving conjugacy class sums and algebraic integers.