Is there a field $F$ such that $F \not \cong F(X)$ as fields but $F(X) \cong F(X,Y)$ ?

An example might not be obvious to find, since it would give an exemple of non-isomorphic fields $K \not\cong L$ such that $K(t) \cong L(t)$ (let $K=F, L=F(X)$). Such an example is given here.

Notice that my question is different from this one (and this one), since there I was asking for $ F(X) \not \cong F \cong F(X,Y) $. Possibly related: (1).

By the way, a similar question for polynomial rings can be asked: is there a unital commutative ring $R$ such that $R \not \cong R[X] \cong R[X,Y]$ (as rings)? I don't have an example either for this one.


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