Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(x_0,x_1,\dots)$. But in this case we have $F \cong F(x)$.
This is related to this question on MO. However, it doesn't follow obviously from $R \not \cong R[x]$ that $\text{Frac}(R) \not \cong \text{Frac}(R)(X)$ (I'm not even sure that the given $R$ is an integral domain...).
These questions could be relevant: (1) ; (2).
Thank you very much!