  A classical theorem of John Thompson on character degrees asserts that if the
degree of every ordinary irreducible character of a finite group $G$ is 1 or
divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a
significant improvement of this result by considering the average of
$p'$-degrees of irreducible characters. We also consider fields of character
values and prove several improvements of earlier related results.
