I wondered the relationship between $Cl_K$ and $Cl_F$ where $K/F$ is extension of number field.
Then I found a following short paper: Hiroyuki OSADA "Note on the ideal class group of abelian number fields".
I have two questions about this:
1.(in the proof of the theorem)
How can we get the map $Gal(\tilde{L} /L)\to Gal(\tilde{K}/K)$? I know it finally equal to composition of norm map and Artin map as wrote below though.
2.(in the proof of the lemma)
Why $f((1-\sigma)x)=0$? I'm also not sure how the action of $G$ defined (it's trivial action as a result though).
Let $L/K/F$ be a tower of Galois extensions. For $\tau \in $ Gal$(K/F)$, let $T$ be a lift to Gal$(L/F)$, i.e., any $F$-automorphism of $L$ that restricts to $\tau$. Then Gal$(K/F)$ acts on Gal$(L/F)$ by conjugation: $\sigma^\tau := T^{-1} \sigma T$. If $L/F$ is abelian, this action is, of course, trivial.
BTW: In such situations, I often find it easier to rewrite the proof in my own words.