Let G G be a profinite group. The purpose of this note is to construct subfields F of the field of complex numbers over which a given finite dimensional complex continuous linear representation D of G G is realizable in the following sense: There is a one-dimensional representation of G G such that m D is isomorphic to a linear representation which is realizable over F. It turns out that certain invariants related to the torsion part of the maximal profinite abelian quotient G G ab of G G play a key role. ᮊ 1998 Academic Press
RATIONALITY OF REPRESENTATIONS
We consider linear and projective representations of finite degree of a profinite group G G and assume throughout that these representations are Ž . matrix representations resp., projective matrix representations over subfields of the field of complex numbers ރ and are continuous with respect Ž to the Krull topology on G G and the discrete topology on the general resp., . the projective linear group. Two such linear representations D , D of G G 1 2 are said to belong to the same genus if there is a linear character Ž . s representation of degree 1 of G G such that D is isomorphic to 2 m D . In this way the set of linear representations of G G splits into 1 equivalence classes which we call genera. The genus of a linear representation D: G G ª GL n, ރ Ž .