A finite extension with finitely many intermediate fields is simple

It is well-known that is $E$ is a simple finite extension over $F$, then it has finite many intermediate fields.

Actually Minle's notes says that if an finite extension has finite intermediate fields, then it is simple! I am curious about this fact, could someone give a proof? Thanks!

Solution 1:

Let $E/F$ is a finite extension (say $[E:F]=n$). If $F$ is a finite field, the result is fairly trivial, so assume $F$ is infinite.

At this point, it is clear that we need only check the case $E=F(a,b)$ is simple i.e. $n=2$, and we can conclude the general result inductively.

Now consider all field extensions of $F$ of the form $F(ax+b)$, for $x \in F$. Since $F$ must have an infinite number of elements, there are infinitely many choices for $x$. Since $F$ has only finitely many intermediate fields, there exists $x\neq y \text{ in } F$ such that $F(ax+b)=F(ay+b)$. Set $c=ax+b$. Since $F(c) =F(ax+b)\subseteq F(a,b)$, it suffices to show $a,b \in F(c)$. Since $x\neq y$ and $ay+b \in F(c)$, we have $$a=\frac{a(x-y)}{x-y}=\frac{(ax+b)-(ay+b)}{x-y} \in F(c)$$

Similarly, $b=c-ax\in F(c)$, so therefore $$F(c)=F(a,b)$$