Let T be a complex irreducible
representation of a finite group G of order n and let χ be the character afforded by T.
An algebraic number field K ⊃ Q(χ) is a splitting field for χ if T can be written in
K. The minimum of [K : Q(χ)], taken over all splitting fields K of χ, is the Schur
index mQ(χ) of χ. In view of the famous theorem of R. Brauer that Q(e2πi∕n) is a
splitting field for χ, it is natural to ask whether there exists a splitting field L with
Q(e2tti∕n) ⊃ L ⊃ Q(χ) and [L : Q(χ)] = mQ(χ). In this paper examples are
constructed which show that such a splitting field L does not always exist. Sufficient
conditions are also obtained which guarantee the existence of a splitting field L as
above.
