Related to a previous [question](https://mathoverflow.net/questions/10126/reference-for-this-theorem-in-representation-theory) 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](http://www.math.fsu.edu/~aluffi/eprint.archive.html).


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 $\chi$?



Faithful characters of finite groups 
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 $\chi$.


