Let $k$ be any field and $K:=k(X_1,X_2,\dots)$ be the field of rational functions over $k$ in countably many variables. Now $K$ has the interesting property that it is isomorphic to a transcendental extension of itself namely $K\cong K(X)$.
Are there any other examples of this phenomenon or is the following true?
When $K$ is a field that is isomorphic to a transcendental extension of itself, then there is some field $k$ s.t. $K\cong k(X_1,X_2,\dots)$.
Best Answer
As written, no.
$\mathbb C$ is isomorphic to $\overline{\mathbb C(X)}$ which is a transcendental extension, but $\mathbb C$ is not isomorphic to anything of the form $k(X_1,X_2,\ldots)$ because the latter is not algebraically closed ($X_1$ does not have a square root, for example).