Why is there a "maximal" intermediate field?

Question

It seems from this answer that the following is true (possibly without the finiteness assumption, I have no idea):

If $E/F$ is a finite field extension, then there exists a maximal intermediate field $E'$.

Presumably this means that $E'$ is an intermediate field and if $K$ is an intermediate field and $E' \subset K$, then $K = E$ (am I right?)

How to prove this?

Answer 1

The correct statement is:

Let $E/F$ be a finite field extension such that $E\neq F$. Then there is a field $E'$ such that $F\subseteq E'\subset E$ and there is no field $K$ such that $E'\subset K\subset E$.

To prove it, consider the set $S$ of all degrees $[E':F]$ of intermediate fields $E'\subset E$. Then $S$ is a nonempty set of natural numbers, all of which are less than $[E:F]$. Let $n$ be the greatest element of $S$, and choose $E'\subset E$ such that $[E':F]=n$. Then if $E'\subset K\subset E$ we would have $$[K:F]=[K:E'][E':F]>[E':F]=n,$$ which is a contradiction since $n$ is the greatest element of $S$.