Characters of π’-degree and small cyclotomic fields

We show that every finite group of order divisible by 2 or q, where q is a prime number, admits a { 2 , q} ′-degree nontrivial irreducible character with values in Q(e2πi/q). We further characterize when such character can be chosen with only rational values in solvable groups. These results follow from more general considerations on groups admitting a { p, q} ′-degree nontrivial irreducible character with values in Q(e2πi/p) or Q(e2πi/q) , for any pair of primes p and q. Along the way, we completely describe simple alternating groups admitting a { p, q} ′-degree nontrivial irreducible character with rational values.