Faithful characters of finite groups  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 $\chi$?
 A: Here is a short proof of the weaker version of the statement from Question 1 (giving a polynomial with rational coefficients). Let's think of characters as functions on conjugacy classes. Then $\chi(1)=n={\rm dim}(V)$, and $\chi(g)$ for $g\ne 1$ has smaller absolute value than $n$ (since the representation is faithful and eigenvalues of $g$ in $\chi$ are roots of 1). In particular, $\chi(g)\ne n$. Now let P be the interpolation polynomial such that $P(n)=|G|$ and $P(x)=0$ for any other value $x$ of $\chi$. Then $P(\chi)$ is the regular character, and it's easy to see that $P$ has rational coefficients.
However, there seems to be a counterexample to the statement that $P$ can be chosen to have integer coefficients. Namely, take $G=A_5$, and $\chi$ the 5-dimensional character. 
Its values are well known to be $5,0,1,-1$, so we can take $P_0=(x^3-x)/2$, and any other 
polynomial which works will be of the form $P=P_0Q$, where $Q$ is another polynomial (as $P$ must vanish at $0,1,-1$). If $P$ has integer coefficients, then $Q/2=P/(x^3-x)$ must have integer coefficients, so values of $Q$ at integers are even. On the other hand, we must have $Q(5)=1$, contradiction.  
A: 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. 
A: I assume that by "faithful irreducible character" you mean the character of a faithful (i.e., trivial kernel) irreducible representation. In this case, the answer to Question 2 is negative. For instance, the irreducible character $\chi$ of the symmetric group $S_4$ indexed by the partition (3,1) is faithful and has the same value on two different conjugacy classes of $S_4$, so the same will be true of any complex polynomial in $\chi$.
