Ideal classes fixed by the Galois group Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any results giving a formula for the number of orbits of this action (or equivalently a formula for the number of ideal classes that are fixed by some element of $G$). In particular, I would like to compare the number of orbits to the class number of $K$.
 A: As Franz Lemmermeyer suggested, one should consider the so-called Ambigous Class Number Formula: you find it, for instance, in Gras' book "Class Field Theory", II.6.2.3.
It says that if $L/K$ is a finite cyclic Galois extension with group $G$ and if we denote by $Cl_L,E_L$ the class group and the unit group of $L$ respectively (and likewise for $K$), then
$$
\vert Cl_L^G\vert=\frac{\vert Cl_K\vert\cdot Ram(L/K)}{[L:K]\cdot [E_K:E_K\cap\mathrm{Norm}_{L/K}(L^\times)]}
$$
where $Ram(L/K)$ is the product of all ramification indexes of finite and infinite primes of $K$ in the extension $L/K$.
So, in case $K=\mathbb{Q}$, you get simply
$$
\vert Cl_L^G\vert=\frac{Ram(L/\mathbb{Q})}{[L:\mathbb{Q}]}\quad\text{or}\quad \vert Cl_L^G\vert=\frac{Ram(L/\mathbb{Q})}{2[L:\mathbb{Q}]}
$$
according as whether $L^\times$ contains or not an element of norm $-1$.
The proof is basically cohomological following the one suggested by Joe Silverman: the term $Ram(L/K)$ comes from fixed ideals in $I_L^G$ and the index of the norm of units of $L$ inside those of $K$ controls some capitulation kernel - but you can look in Gras' book for details.
In case of dihedral extensions, you can look at the question Class groups in dihedral extensions - some sort of Spiegelungssatz?
A: I assume you want $K$ to be Galois over $\mathbb{Q}$. More generally, let $L/K$ be a Galois extension of number fields. The the class group $C_K$ of $K$ maps to $C_L^{G_{L/K}}$, the part of $C_L$ fixed by the Galois group of $L/K$, and you seem to be asking what the quotient $C_L^{G_{L/K}}/C_K$ looks like.
Taking cohomology of the exact sequences
$$ 1\to R_L^*\to L^*\to L^*/R_L*\to1
\quad\text{and}\quad
  1\to L^*/R_L* \to I_L \to C_L \to 1
$$
gives (if I'm not mistaken) exact sequences
$$
  0 \to H^1(G_{L/K},L^*/R_L*) \to H^2(G_{L/K},R_L^*) \to \text{Br}(L/K)
$$
and
$$
  0 \to C_K \to C_L^{G_{L/K}} \to H^1(G_{L/K},L^*/R_L*),
$$
so the quotient that you're interested in naturally injects
$$
  C_L^{G_{L/K}}/C_K \hookrightarrow 
  \text{Ker}\Bigl(H^2(G_{L/K},R_L^*) \to \text{Br}(L/K)\Bigr).
$$
The Galois structure of unit groups has been much studied. You might look at some of Ted Chinburg's papers (http://www.math.upenn.edu/~ted/CVPubs9-10-07.html)
A: Since the map $j$ from $C_K$ to $C_L^{\operatorname{Gal}(L/K)}$ is not always injective, we are looking at the cokernel of this map. $\operatorname{coker} j$ will map onto $$H^1(\operatorname{Gal}(L/K),L^\times/R_{L^\times})= \ker\left(H^2(\operatorname{Gal}(L/K),R_{L^\times}) \to \operatorname{Br}(L/K)\right)$$ with kernel equal to the natural image of $H^1(\operatorname{Gal}(L/K),U_L)$ in $\operatorname{coker} j$, $U_L$ the idele units. In particular we have an isomorphism when $L/K$ is unramified. 
The ambiguous class number formula applies only to cyclic extensions $L/K$.
