Does $K(x)=E(x)$ imply $K=E$?

Question

Let us suppose we have two fields $K$ and $E$ and $K\subseteq E$. Is it true that $K(x)=E(x)$ implies $K=E$? I know it seems sort of obvious, but I don't know if it is actually true. It is for a step in something I'm trying to prove for my homework.

Answer 1

The answer is negative, take $K=\mathbb{Q}$, $E=\mathbb{Q}(x^2)$. Then obviously $K(x)=E(x)$ but $K\neq E$.