Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$.

Question

Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$.

I have tried this:

Suppose that $K \subseteq E \subseteq K(X)$. Let $\alpha \in K$ and $\beta \in E$, then there are $F(\alpha) \in K[X]$ and $G(\beta) \in E[Y]$ such that $K(\alpha) \in K[X]$ and $K(\beta) \in E(Y)$, I don't see any progress with this. Any hint, please?

Answer 1

For $X\notin K$,

if $X^2\notin K$, then $K\subsetneq K(X^2)\subsetneq K(X)$.

If $X^2\in K$, then $X^3\notin K$. (Otherwise, if $X^3\in K$, then $X=\frac{X^3}{X^2}\in K$. $\Rightarrow \Leftarrow$.) Then, $K\subsetneq K(X^3)\subsetneq K(X)$.