I was wondering if someone knows a simpler proof for this fact.

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 $\chi$?

I was wondering if someone knows a simpler proof for this fact.

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 $\chi$?