If degree of extension is infinite then intermediate ring not need to be a field.

Question

Let $F\subset K$ be a field extension and $D$ be an intermediate ring such that $F\subset D\subset K$. If $[K:F]$ is infinite then $D$ is not necessary a field.

So basically I need a counter example. I know this is not true if the degree of extension is finite.

Answer 1

Take $\mathbb{Q} \subset \mathbb{Q}[X] \subset \mathbb{Q}(X)$