# Relative number field and class groups

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".

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).

• (1) is the translation theorem of Galois theory. Clearly $L \overline{K}$ is a subextension of $\overline{L}/L$. (2) G acts by conjugation with some lift; but the group is abelian, so conjugation is trivial.
– user23365
Commented Oct 28, 2022 at 13:04
• @franz lemmermeyer (1) OK, it's merely restriction map isn't it. (2) Sorry, I still don't figure out yet. Do group cohomology or something relate? Commented Oct 31, 2022 at 13:12

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.