Characters, Brauer characters, and local Brauer groups

Let p be a prime. Let G be a finite group, and let χ be an irreducible character of G. Suppose F is a finite extension of Qp, the field of p-adic numbers. If all the values of χ are in F, then, associated with χ, is a specific element of Br(F) the Brauer group of F, and, since Br(F) is canonically isomorphic to Q/Z, χ has, uniquely associated with it, an element of Q/Z. Some recent conjectures and results on modular representation theory depend on the value of this element of Br(F). When χ takes complex values, the element of Br(F) associated with χ is not, in general, uniquely determined. In this paper, we show that, whenever p-Brauer characters are defined, then the element of Br(F) associated with χ can be naturally and uniquely defined.