Let $K$ be a localfield and $O_K$be its ring of integers, and $L/K$ be a quadratic extension.
It is known that $H^1(\text{Gal}(L/K), L) = 0$ according to Hilbert's Theorem 90. However, what is known about $H^1(\text{Gal}(L/K), O_L)$?
Can this be easily deduced from Hilbert's Theorem 90, or is it essentially unrelated?
Indivisual solution for $H^1(\text{Gal}(L/K), O_L) = 0$ is also welcomed. Thank you for your help.
Global case: Relation between Galois Cohomology $H^1(\text{Gal(L/K)}, L) = 0$ and $H^1(\text{Gal}(L/K), O_L)=0$
P.S. What is the essential difference from Hilbert 90? Similarly to Hilbert 90, if we arbitrarily take a cocycle, doesn't it become a coboundary due to the cocycle condition?
This is only a partial answer: I just sketch some reasons of the examples I mentioned in the comment of this question.
See, for example Faltings' paper Faltings' paper. Particularly section 2c. Galois cohomology. or Section 3 of Olsson's article
We have ethe following fact:
Let $G$ be a finite group, $A$ is a commutative ring (with unit), $B$ an $A$-algebra with $G$-action and $A\to B$ is a almost etale covering (see section 2 of Faltings' paper for definition). Let $M$ be a $B$-module with semilinear $G$-action. Then $H^p(G,M)$ is almost zero (see section 1 of Faltings' paper for definition): the idea is that the cohomology group is killed by elements in $A$ of the form $Tr(B):=\{a=\sum_{g\in G}g(b), b\in B\}$, and the "almost etale covering" implies that the image of trace is very big. (killed by something "big" means it was very "small".)
Take $A\to B$ as $O_K\to O_L$ in your case: then if $L/K$ is somehow "almost etale" will imply $Tr(O_L)$ contain all positive valuation elements of $O_K$, including $p$.) (Sees for example a lecture note for some details of this phenomenon.) Hence in particular $H^i(Gal(L/K), O_L)$ will be almost zero, and invert $p$ it tells $H^i(Gal(L/K), L)=0$.
Last few words about those examples I mentioned related with perfectoid fields. Denote $C=\widehat{\overline{\mathbf{Q}_p}}$, we consider for example $K=\mathbf{Q}_p(\zeta_{p^{\infty}})$, then the observation is that (see corollary 185 another lecture notes) the trace image $Tr(O_L)$ for any finite extension $L/K$ is big (contains $\mathfrak{m}_L$) and hence $H^i(Gal(L/K), O_L)$ will be almost zero. (as you can see, from the same lecture note theorem 183, this should be a corollary of the "Almost purity for perfectoid fields", in other words some how from the fact as I phrased "$\widehat{K}$ is a perfectoid field in $C$".)
What is left is then some details and somehow "passing to the limit", for example use the observation $H^i(G_K, O_{\overline{K}} /p^mO_{\overline{K}}) = \varinjlim_{L} H^i(Gal(L/K), O_{L}/p^m O_{L})$ to finally conclude what I said $H^1(G_K, O_C)$ is almost zero.
(Notice also that in the concrete examples I gave, $C$ is not a "local field" (its valuation is not discrete), though we are in the $p$-adic world.)