Can a field be isomorphic to its subfield but not to a subfield in between?

Question

A question related to this one Can a field be isomorphic to its subfield?: are there field extensions K/E and E/F such that K and F are isomorphic but E is not isomorphic to them?

Answer 1

We construct an example. There are many.

Let $t_0,t_1,t_2,t_3,\dots$ be a countably infinite collection of algebraically independent transcendentals.

Let $F$ be the algebraic closure (in $\mathbb{C}$) of $\mathbb{Q}(t_1,t_2,t_3,\dots)$ and let $K$ be the algebraic closure of $\mathbb{Q}(t_0,t_1,t_2, t_3,\dots)$. Let $E=F(t_0)$.