  For various nonsolvable groups $G$, we prove the existence of extensions of
the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order
dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set
for $G$. For these groups $G$, this gives the existence of number fields of
degree $ge(G)$ with an unramified $G$-extension. The existence of such
extensions over $\mathbb{Q}$ for all finite groups would imply that, for every
finite group $G$, there exists a quadratic number field admitting an unramified
$G$-extension, as was recently conjectured. We also provide further evidence
for the existence of such extensions for all finite groups, by proving their
existence when $\mathbb{Q}$ is replaced with a function field $k(t)$ where $k$
is an ample field.
