Field extension with infinite degree

Question

I'm currently working on the proof of the following statement:

Let $L$ be a field and $K$ and $F$ be two subfields such that $F\subseteq K\subseteq L$. Then the following holds for the field extensions $L/F, L/K$ and $K/F$:

$$L/F \ \ \text{is infinite iff.} \ \ L/K \ \ \text{is infinite or} \ \ K/F \ \ \text{is infinite}$$

I've managed to prove the implication from left to right, and, from right to left in the case of $K/F$ is infinite.

How do I prove the other case, i.e. that if $L/K$ is infinite then so is $L/F$?

Answer 1

Suppose that $L/K$ is infinite but $L/F$ is finite.

Then $L$ admits a finite set of generators as a $F$-vector space. These generarators would be also a set of generators as a $K$-vector space, since any linear combination with coefficients in $F$ is also a linear combination with coefficients in $K$. This contradicts the assumption.

Answer 2

The statement is equivalent to its contrapositive

if $L/K$ is finite and $K/F$ is finite, then $L/F$ is finite.

Let $\{x_1,\dots,x_m\}$ be a spanning set of $K$ as a vector space over $F$ and $\{y_1,\dots,y_n\}$ a spanning set of $L$ as a vector space over $L$.

Now consider $\{x_iy_j:1\le i\le m,1\le j\le n\}$ and conclude.

Hint: if $z\in L$, then $$ z=\sum_{j=1}^n \alpha_jy_j $$ with $\alpha_1,\dots,\alpha_n\in K$. Then…

For the converse, you want to prove that if $L/F$ is finite, then both $L/K$ and $K/F$ are finite. If you have a finite spanning set of $L$ as a vector space over $F$, then it is also a spanning set of $L$ as a vector space over $K$; moreover $K$ is a subspace of $L$ (both as vector spaces over $F$).