If we have a field $K$ such that $K\cong K(t)$ (i.e. it is isomorphic to the field you get if you adjoin one transcendental) then is there necessarily a subfield $L\lt K$ such that $L\ncong L(t)$ and $K$ is a purely transcendental extension of $L$?

The basic idea is that if we have something like $K={\mathbb Q}(u_i,t_i^{1/2^n})_{i,n\in\mathbb N}$ then we are looking for the field ${\mathbb Q}(t_i^{1/2^n})_{i,n\in{\mathbb N}}$.