  Let $L/K$ be a Galois extension of number fields. We prove two lower bounds
on the maximum of the degrees of the irreducible complex representations of
${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture
and the Generalized Riemann Hypothesis. Our bound is nontrivial when $[K :
\mathbb{Q}]$ is small and $L$ has small root discriminant, and might be
summarized as saying that such fields can't be "too abelian."
