Fields of values of odd-degree irreducible characters

Abstract

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.