A Note on Rationality of Representations of Profinite Groups

Let be a profinite group. The purpose of this note is to construct subfieldsFof the field of complex numbers over which a given finite dimensional complex continuous linear representationDof is realizable in the following sense: There is a one-dimensional representation λ of such that λ Dis isomorphic to a linear representation which is realizable overF. It turns out that certain invariants related to the torsion part of the maximal profinite abelian quotient abof play a key role.