What can we say about a field $F$ such that any finite extension $K/F$ is Galois?
Clearly, $F$ is perfect. For instance, it seems to hold if $F$ is quasi-finite or $[\overline F : F] < \infty$. What are other examples? (What properties does $F$ have?)
The only thing I can say is that $F(\sqrt[n]{a})$ should contain all the $n$-th roots of unity, since $F(\sqrt[n]{a}) / F$ is Galois, for every $a \in F, n \geq 1$.