I just was looking at an exercise which asks the reader to show that for $F \subset L$, if $L = F(\theta)$ for some $\theta \in L$ then there exists only finitely many subfields $K$ of $L$ containing $F$. Can I just prove this by the following argument:
The tower extension theorem says $[L:F] = [L:K][K:F] $
This means: $[L:F] > [K:F] > 0$ , since $[L:K] = 1 \iff L = K$, which is clear when viewing the extensions as vector spaces.
Continuing to find distinct intermediate subfields $F \subset K' \subset K \subset L $, we see that the degree of the field extension decreases at each step and is bounded below by 0, so the process will eventually terminate.
So there can be at most $[L:K] - 1$ distinct intermediate subfields, which is finite.
Is this a valid argument or is there some subtlety I'm missing here.
Thank you in advance.