Let
be a Galois extension of number fields. We prove two lower bounds on the
maximum of the degrees of the irreducible complex representations of
,
the sharper of which is conditional on the Artin conjecture and the
generalized Riemann hypothesis. Our bound is nontrivial when
is small
and
has small root discriminant, and might be summarized as saying that such fields
can’t be “too abelian”.