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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?

  1. 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 $\mathrm{Gal}(M/L) \rtimes_f \mathrm{Gal}(L/K)$.
  2. What if both intermediate Galois groups are abelian, does this change anything meaningfully?
user10039910
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    In general $M/K$ is not Galois and thus the meaning of $\operatorname{Gal}(M/K)$ is unclear, in particular it may not be a group. Therefore the exact sequence you write doesn't make sense. – WhatsUp Jun 14 '22 at 21:26
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    Given that any short exact sequence of finite abelian groups can occur as a short exact sequence of Galois groups, I doubt your question has a good answer in terms of Galois theory, beyond standard group theoretic criteria. In particular, even for degree 4 extensions, it’s not completely trivial to differentiate between $C_2^2$ and $C_4$ extensions. – Mathmo123 Jun 15 '22 at 12:05
  • Related: https://math.stackexchange.com/q/325671/96384 – Torsten Schoeneberg Feb 12 '25 at 01:35

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