TITLE: A field has only one isomorphic subfield to itself?
QUESTION [8 upvotes]: Let $E$ be a field and $F$ be a subfield of $E$ which is isomorphic to $E$.
Then is $F$ equal to $E$? It seems to be clear but I couldn't prove it.
Could you please explain this statement?

REPLY [11 votes]: Counterexample
Consider $\mathbb{R}$ and consider the fields of rational functions $\mathbb{R}(x^2)$ and $\mathbb{R}(x)$. They are isomorphic but $\mathbb{R}(x^2) \subset \mathbb{R}(x)$.