Irreducible character of degree greater than one takes value zero on some conjugacy class

Statement
Suppose $$G$$ is a finite group and $$\chi$$ is the character of an irreducible linear representation of $$G$$ over $$\mathbb{C}$$, such that the degree of the representation (and hence, of $$\chi$$) is greater than one. Then, there exists an element $$g \in G$$ (and hence, a fact about::conjugacy class) such that $$\chi(g) = 0$$.

Related facts

 * Zero-or-scalar lemma
 * Conjugacy class of more than average size has character value zero for some irreducible character

Facts used

 * 1) uses::Character orthogonality theorem
 * 2) uses::Sufficiently large implies splitting: In particular, this exhibits a splitting field of characteristic zero that is a finite cyclotomic extension of the rationals.
 * 3) Cauchy-Schwartz inequality
 * 4) uses::Characters are algebraic integers

Proof
Given: A finite group $$G$$, $$\chi$$ the character of an irreducible representation of degree greater than $$1$$ of the group $$G$$. $$e$$ is the identity element, $$d = \chi(e)$$ is the degree, so $$d > 1$$.

To prove: There exists $$g \in G \setminus \{ e \}$$ such that $$\chi(g) = 0$$.

Proof: