gr.group theory - Faithful characters of finite groups

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 $gne 1$ 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.