Fields of values of odd-degree irreducible characters

In this paper we clarify the quadratic irrationalities that can be admitted by an odd-degree complex irreducible character χ of an arbitrary finite group. Write Q(χ) to denote the field generated over the rational numbers by the values of χ, and let d > 1 be a square-free integer. We prove that if Q(χ) = Q( √ d) then d ≡ 1 (mod 4) and if Q(χ) = Q( √ −d), then d ≡ 3 (mod 4). This follows from the main result of this paper: either i ∈ Q(χ) or Q(χ) ⊆ Q(exp(2πi/m)) for some odd integer m ≥ 1