Gal$(K/\mathbb{Q})$ invariants of $Cl(K)$. Let $K$ be a finite Galois extension of $\mathbb{Q}$, with $G$ its Galois group. $G$ acts on the group $Div(K)$ and preserves the subgroup $Prin(K)$ of principal divisors, thus the action descends to the quotient $Cl(K)$. What do the subgroup of classes invariant under $G$ look like? Can we use the cohomology of groups to establish this?
If it is not known (or difficult) in general, what are some simple, instructive cases?
 A: As pointed out by F. Lemmermeyer in the link  mathoverflow.net/questions/177408/… given by Watson, your question is at the heart of the subject called "genus theory" in ANT, under the headlines "capitulation" and "ambiguous class number formula". Here is the general setting : given a finite Galois extension $K/k$ with group $G$, we propose to investigate the natural (sometimes called capitulation) map $j: Cl_k \to {Cl_K}^G$ via the study of its kernel $Cap(K/k)$ and cokernel $Cocap (K/k)$. The multiplicity and diversity of existing results (see e.g. the bibliography of the book [G], chap. IV) could give a false impression of confusion and make us lose sight of the fact that behind the scene, it is CFT which gives their unity to the different approaches. Of course CFT admits many formulations - using ideals, idèles, cohomology -  which can be selected according to your chosen goal. Here you ask for a cohomological approach, but even with this narrow targeting, it is necessary to follow some guidelines. 
Notations.  $O_K$ = the ring of integers of $K$, $E_K$ = the group of units, $I_K$ = the group of ideals, $P_K= K^* /E_K$ = the group of principal ideals , $Cl_K=I_K /P_K$ = the ideal class group, $J_K$ = the group of idèles, $C_K$ = the group of idèle classes, $U_K$ = the group of unit idèles, i.e. idèles of $K$ whose components are all local units (recall that by convention the unit group at an archimedean place is $\mathbf R^*$ or $\mathbf C^*$).
Let us choose to concentrate on the "resolution" of $Cap(K/k)$ and $Cocap (K/k)$ by means of exact sequences. Without any pretention at exhaustivity, here are a few typical examples (where $G$ is omitted in the cohomology groups) :
(DSR) (Dok Sang Rim 1965) $0\to Cap(K/k) \to H^1 (E_K) \to {I_K}^G/I_k \to Cocap(K/k) \to H^2 (E_K) \to Ker (H^2 (K^*)\to H^2 (I_K))\to H^1(Cl_K)\to H^3(E_K)$ 
(Iw) (Iwasawa 1983) $0\to H^1 (E_K) \to {I_K}^G/P_k \to {Cl_K}^G\to H^2 (E_K) \to Ker (H^2 (K^*)\to H^2 (I_K))\to H^1(Cl_K)\to H^3(E_K)$ 
NB. If $G$ is cyclic, the map going to $H^1(Cl_K)$ at the right of both sequences is surjective
(KL) (Karoubi & Lambre 2007) $0 \to (E_K)^{N=1} \to (K^*)^{N=1}\times (U_K)^{N=1} \to (J_K)^{N=1} \to \mathbf K_0(N) \to \hat H^0(K^*)\times
\hat H^0(U_K) \to \hat H^0(J_K) \to Cl_k /N(Cl_K)$  
where $N$ is the norm map of $K/k$, $(.)^{N=1}$ is the kernel of the norm, and $\mathbf K_0(N)$ denotes the $\mathbf K_0$-group of the functor "restriction of scalars" from the category $Proj (O_K)$ of $O_K$-projective noetherian modules to the corresponding category $Proj (O_k)$
All the above were originally obtained by various independent methods, but there is actually a unified approach starting from a classical commutative diagram with 3 exact lines and 3 exact columns relating the objects intoduced in the notations. An immediate consequence is the so called "ambiguous class number formula" in the general case. If $\theta$ denotes the natural map $H^2(E_K)\to H^2(U_K)$ , then $\mid Cl_K^G\mid = \frac {\mid Cl_k \mid}{\mid Im\theta\mid}.\frac {\mid H^2 (E_K)\mid}{\mid H^1 (E_K)\mid}.\prod e_v$, where $e_v$ is the ramification index for $v$ running through the finite places of $k$ which ramify in $K$. In the particular case where $G$ is cyclic, this formula boils down to $\mid Cl_K^G\mid = \frac {\mid Cl_k \mid}{(E_k:E_k\cap NJ_K)}.\frac {\prod e_v}{[K:k]}.2^a$, where $a$ is the number of real places of $k$ which become complex in $K$ (Chevalley 1951).
Sketchy ref.
[G] G. Gras, "Class field theory-From theory to practice", Springer Monographs in Math.
[L1] F. Lemmermeyer, "The ambiguous class number formula revisited"
