Can extension by an isomorphic field be of degree at least 2?

Question

Suppose $K/F$ is a field extension such that $K\not=F$.
Is it legitimate to say that $F$ and $K$ can't be isomorphic since by assumption \begin{equation*}[K:F]\ge 2\end{equation*}and if $K$ and $F$ were indeed isomorphic then we could say that \begin{equation*}[F:F]\ge 2\end{equation*}but $[F:F]$ should be $1$?

Answer 1

In fact, they can be isomorphic.

For example $\mathbb{R}(X^2) \subset \mathbb{R}(X)$ is a field extension of degree $2$ but the two fields are isomorphic (as fields) by $X^2 \mapsto X$.

This isomorphism (as fields) does however not induced an isomorphism of $F$-vectorspaces between $F$ and $K$, which is why $[K:F]\neq [F:F]$ is not a problem even when $K\simeq F$ as fields.