I have some problems concerning the chapter about "The Class Formation of Unramified Extensions" in Neukirch's "Class Field Theory".
$K$ is a $\mathfrak{p}$-adic number field with finite residue field and we only consider unramified finite extensions $L/K$ of $\mathfrak{p}$-adic number fields. Since the extension is unaramified ($e=1$) the unique extension of $v_K$ (the valuation of $K$) is $v_L=\frac{1}{e}v_K=v_K$. Let $G_{L/K}$ be the Galois group of $L/K$.
Now thinking about Tate cohomology groups one gets the result that $H^1(G_{L/K},\mathbb{Q}/\mathbb{Z})=Hom(G_{L/K},\mathbb{Q}/\mathbb{Z})=\chi(G_{L/K})$, the group of characters of $G_{L/K}$, because $\mathbb{Q}/\mathbb{Z}$ acts trivially on $G_{L/K}$. Moreover there is an isomorphism $\overline{v}$ induced by the valuation $v_K$ so that $H^2(G_{L/K},L^\times)\cong H^2(G_{L/K},\mathbb{Z})$. Using the exact sequence $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow0$ and the connection map $\delta$ we get another isomorphism: $H^2(G_{L/K},\mathbb{Z})\stackrel{\delta^{-1}}{\longrightarrow}H^1(G_{L/K},\mathbb{Q}/\mathbb{Z})=\chi(G_{L/K})$. The third important isomorphism is induced by the Frobenius automorphism $\varphi_{L/K}$ with finite order $[L:K]$ which is a generator of $G_{L/K}$: $\chi(G_{L/K})\stackrel{\varphi}{\rightarrow}\frac{1}{[L:K]}\mathbb{Z}/\mathbb{Z}$.
Putting the pieces together one gets the isomorphism $H^2(G_{L/K},L^\times)\stackrel{\overline{v}}{\longrightarrow}H^2(G_{L/K},\mathbb{Z})\stackrel{\delta^{-1}}{\longrightarrow}\chi(G_{L/K})\stackrel{\varphi}{\longrightarrow}\frac{1}{[L:K]}\mathbb{Z}/\mathbb{Z}.$
Now I have two questions:
1) How do I check that $\varphi$ is an isomorphism and that $im(\varphi)\subseteq\frac{1}{[L:K]}\mathbb{Z}/\mathbb{Z}$?
I tried to take a character $\chi\in\chi(G_{L/K})$ and to use the fact that $\varphi_{L/K}^{[L:K]}=id$. Then the following holds: \begin{align}&[L:K]\varphi(\chi)=[L:K]\chi(\varphi_{L/K})=\chi(\varphi_{L/K}^{[L:K]})=\chi(id)\equiv 0\mod\mathbb{Z}\\ \Leftrightarrow&[L:K]\varphi(\chi)\equiv \frac{m\cdot[L:K]}{[L:K]}\mod\mathbb{Z}\\ \Leftrightarrow&\varphi(\chi)\equiv \frac{m}{[L:K]}\mod\mathbb{Z}\end{align} for some $m\in\mathbb{Z}$. Does this make any sense? Now I don't know how to go on.
2) There is an equation in which using the isomorphism $\overline{v}$ causes that a cup product changes to a normal product. I really don't see the reason why:
Let $\delta_\chi\in H^2(G_{L/K},\mathbb{Z})$ be the image of $\chi\in\chi(G_{L/K})$ under $\delta$ and $\overline{a}=a~N_{L/K}(L^\times)\in H^0(G_{L/K},K^\times)=K^\times/N_{L/K}(L^\times)$ with $a\in K^\times$.
I want to check why $\overline{v}(\overline{a}\cup\delta_\chi)=v_K(a)\cdot \delta_\chi$.
My idea is the following: $\overline{v}(\overline{a}\cup\delta_\chi)=\overline{v_K(a)\cup v_K(\delta_\chi)}=\overline{v_K(a)\cup \delta_\chi}$ because $\delta_\chi$ already is in $\mathbb{Z}$.