TITLE: Faithful characters of finite groups 
QUESTION [8 upvotes]: Related to a previous question I am asking furthermore a proof
for the following:
Question 1: If $\chi$ is a faithful irreducible character of a
finite group $G$ then the regular character of $G$ is a polynomial
with integer coefficients in $\chi$?
I know this fact is true since there is a generalization of it for
Hopf algebras in Corollary 19 of the paper FSU96-08 from here.
The proof from that paper is a little complicated using some
(although elementary) results on norms and inner products.
I was wondering if  anyone knows a different proof of this.
Using the Stone - Weierstrass method mentioned in the previous
question, I am asking further if the following is true:
Question 2: If $\chi$ is a faithful irreducible character of a
finite group $G$ does any character of $G$ is a complex polynomial
in $\chi$?

REPLY [5 votes]: I put an answer (due to Blichfeldt, not me) to essentially this question
at your earlier question. To address the problem raised by Richard Stanley,
one result I know in this direction is by John Thompson: if $\chi$ is an irreducible
character of a finite group $G$, then there are more than $|G|/3$ elements at which 
the value taken by $\chi$ is either zero or a root of unity.