Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.


Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.


Can $L$ contain a Noetherian unital subring $R\subsetneq L$ not contained in any subfield $F\subsetneq L$?



Non-abelian isomorphic absolute Galois groups of fields of different characteristic
I'm not sure I understand the point of the question, but the answer is yes: For example, $K=\overline{\mathbb{F}\_p}(t)$ and $L=\overline{\mathbb{Q}}(t)$ are both known to have absolute Galois group free profinite on countably many generators, but $R=\overline{\mathbb{Q}}[t]$ is a noetherian proper subring of $L$ with quotient field $L$.


