I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$. And I have a lot of questions to this chapter. In this question I will start with an easy question. Let us start with:
My question : So I don't understand (a) and (b). Like you can see we need Proposition 11.2.2. So here:
Like I said: can you proof (a) and (b)? ( not the (a) and (b) from the Proposition. I mean the (a) and (b) of the first picture. ) It is not clear to me. I solved (c) by my own, I think:
$ \chi^{'}(a) = \chi(N_{F_s/F}(a)) = \chi(a^s * N_{F_s/F}(1)) = \chi(a^s) = (\chi(a))^s $ I used Proposition 11.2.2 (c) for the second equation.
So if you need more information let me know.
Let us start with (a): So you know that $ \chi \neq \rho $. That means that there is an element $ a \in F^*$ with $\chi(a) \neq \rho(a)$.
We use Proposition 11.2.2 (d). That means that there is an element $ \beta \in F_s^* $ with $N_{F_s/F}(\beta)= a$
We get with the help of Proposition 11.2.2(a): $\chi^{'}(\beta) = \chi(N_{F_s/F}(\beta)) \neq \rho(N_{F_s/F}(\beta))=\rho^{'}(\beta). $ So $ \chi^{'} \neq \rho^{'}$.
Now let us solve (b) : Remember $(\chi^{'})^m = \varepsilon$ <=> $(\chi^{'})^m(\alpha) = 1$.
We want to use Proposition 11.2.2(a) again. $\chi^{'}(\alpha)^m = \chi(N_{F_s/F}(\alpha))^m = 1 \mbox{ } \mbox{ } \forall \alpha \in F_s $