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Let $F$, $L$ be finite extensions of a field $K$.

Question: Is the compositum $FL$ a finite extension of $K$?

My motivation: My question actually arised from the proof of Lemma 2 from the paper "Euler Factors determine local Weil Representations" by Tim and Vladimir Dokchitser: Let $K$ be a local field, $F/K$ be a cyclic Galois extension whose degree is a prime power, and $L/K$ is a cyclic totally ramified Galois extension whose degree is the ramification index of $F/K$ with certain properties (cf. paper). I need to show that $FL/L$ has the same degree as $F/K$, and found a proof for $[FL:L]\geq [F:K]$, so $[FL:L]\leq [F:K]$ is the missing inequality left. But I was not even sure if $FL/L$ is finite. If $FL/K$ would be finite, then $FL/L$ would be finite too.

You see that my original question is really specific, so I am not sure how to start. The difficult thing here is that I do not see any relationship between $F$ and $L$. Could you please help me with this problem? Thank you!

Diglett
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    +1 for realizing that there is a problem in talking about $FL$. The compositum is uniquely defined only inside a fixed "umbrella" field $\Omega$ that contains both $F$ and $L$. We can always construct such an umbrella field, but then, in general, $FL$ won't be uniquely determined even up to isomorphism. – Jyrki Lahtonen Oct 23 '18 at 20:54
  • See, for example, this thread. There we answer your question in the affirmative. – Jyrki Lahtonen Oct 23 '18 at 20:55
  • And there is a high chance that the concerns in my first comment need not apply to your case. Those extra conditions will likely come to the fore. – Jyrki Lahtonen Oct 23 '18 at 20:59

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