Primes and conductors of characters

If G is a finite p-solvable group, P is a Sylow p-subgroup of G and χ∈Irr(G) is an irreducible complex character of degree not divisible by p, we prove that the field of values Q(χP) of the restriction of χ to P has index not divisible by p in the cyclotomic extension Q(e2πipf), where pf is the p-part of the conductor of χ. This proves the p-solvable case of the main conjecture in 4.