Quadratic rational solvable groups

A finite group G is quadratic rational if all its irreducible characters are either rational or quadratic. If G is a quadratic rational solvable group, we show that the prime divisors of |G| lie in {2,3,5,7,13}, and no prime can be removed from this list. More generally, if G is solvable and the field Q(χ) generated by the values of χ over Q satisfies |Q(χ):Q|⩽k, for all χ∈Irr(G), then the set of prime divisors of |G| is bounded in terms of k. Also, we prove that the degree of the field generated by the values of all characters of a semi-rational solvable group (see Chillag and Dolfi, 2010 [1]) or a quadratic rational solvable group over Q is bounded, giving a positive answer to a question by D. Chillag and S. Dolfi.