we consider number fields embedded in a fixed algebraic closure $k$ of $\mathbb{Q}$, or simply consider number fields in $\mathbb{C}$. for $L$ a number field, we write $G_L$ for the galois group $Gal(k/L)$.
Take $F$, $F_1$, $F_2$ and $E$ number fields such that $F=F_1\cap F_2$, $E=F_1F_2$, and $F_1$ Galois over $F$. Then $G_E=G_{F_1}\cap G_{F_2}$ and $G_F=G_{F_1}G_{F_2}$ (as we have assumed that $F_1$ is galois over $F$).
Then what happens with the abelianization? the norm maps give us a diagram
$$\xymatrix{ G_E^{ab} \ar[d]^{\subset} \ar[r]^{\subset} & G_{F_1}^{ab} \ar[d]^\subset \ G_{F_2}^{ab} \ar[r]^\subset & G_F^{ab}}$$
Is there chance that this diagram is cartesian?
We can also write $T^L$ for the torus $Res_{L/\mathbb{Q}}\mathbb{G}_\mathrm{m}$. Then the diagram above can be produced from the one below by passing to connected components of idele classes
$$\xymatrix{T^E \ar[d] \ar[r] & T^{F_1} \ar[d] \ T^{F_2} \ar[r] & \T^F}$$
However by dimension arguments one cannot expect this lower diagram to be cartesian. What happens if we only take connected components of the idele class groups?
thanks.
