Hilbert divisors and degrees of irreducible Brauer characters

In this paper, we prove that the Hilbert divisors of irreducible Brauer characters in 2-blocks with nontrivial abelian defect groups are strictly greater than 1. This confirms a conjecture of Liu and Willems in this case. The proof relates the conjecture with a problem of Feit, which asks if the p-part of the degree of an irreducible Brauer character phi of G is always less than the p-part of the order of G. We resolve Feit's problem positively for 2-blocks with abelian defect groups. But it is well known that the question has a negative answer for 2-blocks with non-abelian defect groups.