Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ of $F(X)$ of the same cardinality as $X$ that is not isomorphic to $F(X)$?
I know that standard arguments show that each subfield of $F(X)$ of the same cardinality as $X$ has an isomorphic copy of $F(X)$ as a subfield.
I am also aware of the fact that the "Lüroth problem" has negative solutions for $n=2$ and $n=3$.
PS. This question was initially posed in 2005 on Math Forum, but did not result in any answers.