Let $K$ be a field and $K(X)$ be the field of its rational functions.
Now let $\phi \in K(X)$ be a rational function such that $K(\phi) \neq K(X)$.
Now, since $\phi$ is transcendental over $K$, $K(\phi)$ is isomorphic to $K(X)$.
Is this a correct example of a field being isomorphic to its subfield?
Are there any other examples?