Let $M \supset L \supset K$ be a tower of field extensions where all the extensions are Galois.
There is an exact sequence $$1 \rightarrow \mathrm{Gal}(M/L) \rightarrow \mathrm{Gal}(M/K) \rightarrow \mathrm{Gal}(L/K) \rightarrow 1.$$
My question is, what do we know about $\mathrm{Gal}(M/K)$? At least $\mathrm{Gal}(M/K)$ injects into the wreath product of the $\mathrm{Gal}(M/L)$ with $\mathrm{Gal}(L/K)$, but do we know more?
- Is there a criteria for recognizing $\mathrm{Gal}(M/K)$ as a semi-direct product of the other two Galois groups, i.e. as some $\mathrm{Gal}(M/L) \rtimes \mathrm{Gal}(L/K)$? I can't think of an obvious reason for the sequence above to be split, even if there was a group homomorphism $f\colon\mathrm{Gal}(L/K) \rightarrow \mathrm{Aut}(\mathrm{Gal}(M/L))$ giving rise to a semi-direct product $\mathrm{Gal}(M/L) \rtimes_f \mathrm{Gal}(L/K)$.
- What if both intermediate Galois groups are abelian, does this change anything meaningfully?