Let $L/K$ be algebraic extensions of $\Bbb Q_p$. Consider $L^{\times}$ with the $p$-adic topology. Is it true that the first continuous cohomology group $H^1_{cont}(\mathrm{Gal}(L/K), L^{\times})$ vanishes?
I know that this is true if $L^{\times}$ has the discrete topology. For me, the usual proof works if $L/K$ is finite. Is that correct? In the infinite case, we may not be able to apply 2.2.16 here, however (since this requires to have discrete Galois modules).
On
NB : This is intended to be a comment, but I need the space.
I would say yes for $H^1(G,\bar K^*)=0$, where $G$ is the absolute Galois group of the local $p$-adic field $K$, and $H^1(G,\bar K^*)$ is defined just as in Galois cohomology, i.e. as the direct limit of the finite cohomology groups $H^1(G/H,(\bar K^*)^H)$, where $H$ runs through the normal subgroups of $G$ of finite index (recall that the elements of $G/H$ are then automatically continuous w.r.t. the $p$-adic topology). It remains to verify whether the $H^1$ so defined is a $H_{cont}^1$ w.r.t. the topologies you selected : the topology on $G$ is natural in the Galois context, that on $\bar K^*$comes from extension of the valuations at finite levels, and you must check that the two are compatible with the action of $G$. The answer is probably yes, but I feel too lazy to work on this.
One reason for my laziness is that this question does not seem natural, because the $p$-adic topology on the fields plays no serious role. A better question would be about the $p$-adic completion $C$ of $\bar K$, where new phenomena occur. In the additive version of Hilbert's 90, you get as before $H^1(G,K)=0$, but things change dramatically with $C$. The following results are not at all easy to prove (see Tate, "$p$-divisible groups", Driebergen Proc., 1966) :
1) $H^0(G, C) = K$ ; 2) $H^1 (G, C)$ is a $1$-dimensional $K$-vector space (i.e. the additive version of Hilbert's 90 does not hold)
The much deeper importance of Tate's introduction of his $p$-divisible groups (nowadays better called Barsotti-Tate groups) is their application to $p$-adic Hodge theory, with the classification and study of $p$-adic Galois representations by Fontaine and his followers. Curiously enough, the cohomology of $C^*$ has not given rise to similar developments.
Let $L/K$ be a Galois extension of fields (finite or not), with Galois group $G$. To have a clearer view, we must go back to the definition of the $H^r(G, A), r\ge 0$, attached to a $G$-module $A$. The main goal is not limited to these cohomology groups by themselves, it is to construct right derived functors of the functor "$G$-invariants". More precisely, functors which produce, starting from any short exact sequence of $G$-modules $0\to A \to B \to C \to 0$, a canonical long exact sequence $0\to A^G \to B^G \to B^G \to H^1(G, A)\to H^1(G, B)\to H^1(G, C)\to ...$ Heuristically, you should think of the Taylor series expansion, when it exists, of a function $f$, which approximates a given value, say $f(0)$ , in the neighborhood of $0$. The interest of the process is its automatic character: you don't need to "think" when writing down such an expansion, serious work begins only after, in the interpretation of the coefficients.
0) This being said, the abstract theory of group cohomology doesn't add any constraint on the group $G$ nor the $G$-module $A$. It is standardly constructed using cochains, but there is a uniqueness theorem which ensures that the final result is canonical. The best presentation, I think, can be found in Serre's "Local Fields": generalities (chap. VII), finite groups (chap. VIII,IX), Galois cohomology (chap. X).
1) Your problem belongs to Galois cohomology. The cohomological version of Hilbert's thm.90 states that $H^1(G, L^*)=0$ when $G$ is finite. It's usually proved using Dedekind's theorem on the linear independence of automorphisms. You ask two questions:
i) Does the statement remain valid when $G$ is finite and $L^*$ is a finite extension of $\mathbf Q_p$, endowed with its $p$-adic topology (= defined by the $p$-adic valuation $v_L$) ? Since we deal with $G$-modules $A$, one natural preliminary requirement, when introducing topology, is the continuity of the action of the topological group on the topological module. If $G$ is finite, there is no choice on the topology of $G$, it must be discrete. Here you impose the $p$-adic topology on $L^*$, so the question boils down to the continuity or not of the group action already given together with $L/K$. The answer is positive: any $s\in G$ is continuous because $v_l (s(a))-a) \ge 0$ for all $a\in L^*$ s.t. $v_L(a)\ge 0$ (loc. cit., chap. IV, §1, lemma 1).
ii) What happens when $G$ is infinite ? Because of the very essence of Galois theory, you have no choice on the topology of $G$, it must be the profinite topology. As for $L^*$, again because of Galois theory, a natural choice is the discrete topology. In this setting, called Galois cohomology, you have a cohomological functor in the sense of the recap above, and it works perfectly, as shown by the cohomological version of CFT.
You ask about the $p$-adic topology when $L/K$ is infinite, but this is not natural at all. For example, if $L=K_{nr}$, the maximal unramified extension of $K$, it is known that $G$ is pro-cyclic, topologically generated by the Frobenius automorphism, and in Galois cohomology, as we said, $H^1(G, K_{nr}^*)=0$. But $K_{nr}$ is not complete. To recover the usual advantage of local fields (your motivation?), you could replace $K_{nr}$ by its completion $\hat K_{nr}$ and extend the action of $G$ by continuity. But then, $\hat K_{nr}$ is not a topological $G$-module (loc. cit., chap. XIII, §5, comment after coroll. 2).
3) Of course the "naturalness" of Galois cohomology does not mean that it should be the only interesting kind of continuous cohomology, i.e. cohomology built from continuous cochains. According to the subject, one could drop/restrict one or another property required for a cohomological functor. Examples:
i) The drawkack of continuous cochains is that they give long exact sequences of cohomology groups only for those short exact sequences of modules which are topologically split. But this restriction does not destroy everything, as shown by Tate in his study of the relations between Galois cohomology and K-theory.
ii) Replacing continuous by measurable cochains (Moore) allows to construct a cohomological functor in the category of locally compact groups acting on locally compact modules whose toplogies are induced by a complete separable metric.
iii) In the context of non abelian CFT (Langlands' program), the analogues of the Galois group such as the Weil group or the conjectural Langlands group (whose finite dimensional representations are supposed to parametrize automorphic representations) are no longer profinite but locally compact. The (partly conjectural) Weil-étale cohomology groups proposed by Lichtenbaum are then the right derived functors of the functor of invariants, just as in the recap ./.