let $K$ be an algebraically closed field. Consider the algebraic closure $\overline{K(X)}$ of $K(X)$, with $X$ trascendent over $K$. Are there cases in which $\overline{K(X)}\cong K$? where $\cong$ is isomorphism in whatever sense u prefer. Example: if we consider $K=\mathbb{C}$ then this is true if we take $\cong$ isomorphism of $\mathbb{Q}$ vector spaces. What about field structure?
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