# Galois descent for profinite groups acting on local fields

Suppose that $$G$$ is a profinite group acting faithfully on a field $$L$$ and assume moreover that the action is admissible, that is, that every $$l \in L$$ is stabilized by an open subgroup of $$G$$. If that's the case, then we have Galois descent, that is, there's an equivalence

$$Vect_{L}^{G} \simeq Vect_{L^{G}}$$

between the category of $$L$$-vector spaces equipped with a compatible admissible action of $$G$$, and the category of vector spaces over the fixed field $$L^{G}$$. In particular, the category $$Vect_{L}^{G}$$ is semisimple.

Question: What happens if we remove the admissibility assumption? To give a precise example, suppose that $$G$$ acts faithfully and continuously on a complete discrete valuation ring $$R$$ with a finite residue field $$R / \mathfrak{m} \simeq k$$, where we consider the latter as equipped with the profinite topology coming from the isomorphism $$R \simeq \varprojlim R / \mathfrak{m}^{n}$$. If that's the case, then the action of $$G$$ extends to a continuous action on the fraction field $$K = (R)$$.

Do we have any form of Galois descent in the case of such an action? How is the category of (say, finite-dimensional) $$K$$-vector spaces equipped with a compatible continuous $$G$$-action related to the category of vector spaces over the fixed field $$K^{G}$$? Is the category of such representations necessarily semisimple?