Edit: According to the comment of Mindlack this special question is solved, but yet I have no idea if my calculations lead to another number $r\neq 0$. And in the case $r\neq 0$, I don't know how should I continue.
I have a very simple question about 1-cocycles $\Big( f(\sigma\tau)=\sigma f(\tau)+f(\sigma) \Big)$: Let $G:=Gal(\mathbb{Q}(i)/\mathbb{Q})$, and I denote the identity automorphism by $\sigma_0$. Is my following conclusion correct?
$$f(\sigma_{0}\sigma_{0})=\sigma_{0}f(\sigma_{0})+f(\sigma_0) \Longrightarrow f(\sigma_{0})=\sigma_{0}f(\sigma_{0})+f(\sigma_0) \Longrightarrow \sigma_{0}f(\sigma_{0}) =0 \Longrightarrow f(\sigma_{0}) =0, $$
but $\mathbb{Q}(i)^{\times}$ does not have $0$. What is wrong with my calculations?