$L/K$ field extension and $[M : K] ≤ C$ holds for all real intermediate fields $M$. Then $[L : K]$ is also finite

Question

I want to check my solution for this exercise:

Let $L/K$ be a field extension and there exists a constant $C ∈ \mathbb{N}$ such that $[M : K] ≤ C$ holds for all proper intermediate fields $M$. Then $[L : K]$ is also finite.

To prove this can I not simply do this?:

$L=\cup_{i=0}^n M_i$ for $K\subset M_i \subset L$ $\Rightarrow [L:K]\leq C^n\Rightarrow$ $[L:K]$ is finite.

Answer 1

First note that $L$ is algebraic over $K$: were an element $t \in L \setminus K$ transcendental, then $K\subsetneq K(t^2)$ would be a proper infinite subextension.

The poset of proper intermediate fields (wrt inclusion) can be empty only when $L/K$ is finite, so we can consider the nonempty case. By problem statement the poset has finite height, so we can find a maximal element $L/F/K$. Take $\alpha \in L \setminus F$. Then the compositum $F(\alpha)$ is equal to $L$, and it has $K$-dimension at most $[F:K] [K(\alpha):K] \le C^2$.

Exercise: find a tower $L/F/K$ that attains the equality.