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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 $F\subsetneq L$?

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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 $L$.

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