A month ago I've asked two questions about rationality of the zeta function. The pages that belongs to my question are (linked here) Unfortunately I'm still clueless, but some steps are clear now.
We start with:
$$ \sum_{\chi_0^{(s)},...,\chi_n^{(s)}} \chi_0^{(s)}(a_o^{-1}) \cdots \chi_n^{(s)}(a_n^{-1})g(\chi_0^{(s)}) \cdots g(\chi_n^{(s)}) \mbox{ } \mbox{ } \mbox{ }\mbox{ } \mbox{ } $$
$\chi_i^{(s)}$ are multiplicative characters of $F_s$, such that $\chi_i^{(s)m} = \varepsilon$, $\chi_i^{(s)} \neq \varepsilon$ und $\chi_0^{(s)} \cdots \chi_n^{(s)} = \varepsilon$ .
We showed that $\chi^{'} = \chi \circ N_{F_s/F}$ is a character of $F_s$ with:
(a) $ \chi \neq \rho $ implies that $\chi^{'} \neq \rho^{'} $
(b) $ \chi^{m} = \varepsilon $ implies that $\chi^{'m} = \varepsilon $
(c) $ \chi^{'}(a) = \chi(a)^s $ for all $a \in F$.
with the help of the information above I can almost replace $$ \sum_{\chi_0^{(s)},...,\chi_n^{(s)}} \chi_0^{(s)}(a_o^{-1}) \cdots \chi_n^{(s)}(a_n^{-1})g(\chi_0^{(s)}) \cdots g(\chi_n^{(s)}) \mbox{ } \mbox{ } \mbox{ }\mbox{ } \mbox{ } $$ with $$ \sum_{\chi_0,...,\chi_n} \chi_0(a_o^{-1})^s \cdots \chi_n(a_n^{-1})^sg(\chi_0^{'}) \cdots g(\chi_n^{'})$$ why almost? : I only need to show the part " it follows that as $\chi$ varies over characters of $F$ of order dividing $m$, $\chi^{'}$ varies over characters of $F_s$ of order dividing $m$ ". Please help me here.
If you need more information. Please let me know.