Let $G$ be a finite group and $(V, \sigma)$ be a non-trivial irreducible (complex) representation of $G$. Let $\chi$ denotes the corresponding character and define the polynomial $p(x)=\prod_{g\in G}(x-\chi(g))$ over $\mathbf C$.
Is there anything known about this polynomial? In particular, what can we say about its Galois group? EDIT: It is not even clear to me whether this polynomial always is a rational polynomial (I was tacitly assuming this before). But because of the symmetry it seems that it should be.
All I can see is that the coefficient of $x^{|G|-1}$ is $0$ since it is equal to $|G|$ times the inner product of $\chi$ with the character coming from the trivial representation.
Partial answer. Yes, the constant term of this polynomial is $0$ if $\chi$ is non-linear: a theorem of Burnside states that every non-linear complex character has a zero. For a linear character $\lambda$, one can prove the constant term is $\pm 1$. So you could think one should focus on linear characters. Still then, take a linear integral character, for example the non-principal linear of $S_4$, then $p(x)=(x-1)(x+1)^6(x-1)^3(x-1)^8(x+1)^6=(x^2-1)^{12}$.