Trying to understand why the zeta function is a rational function under certain conditions. Questions about some equations.

Information: I linked the pages below, which relate to my questions.

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 $$a_0x_0^m+a_1x_1^m+...+a_nx_n^m$$. They start with:

$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + \frac{1}{q^s} \sum_{\chi_0^{(s)},...,\chi_n^{(s)}} \chi_0^{(s)}(a_o^{-1}) \cdots \chi_n^{(s)}(a_n^{-1})g(\chi_0^{(s)}) \cdots g(\chi_n^{(s)})$$

That's ok, but I don't see why $$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + \frac{1}{q^s} \sum_{\chi_0^{(s)},...,\chi_n^{(s)}} \chi_0^{(s)}(a_o^{-1}) \cdots \chi_n^{(s)}(a_n^{-1})g(\chi_0^{(s)}) \cdots g(\chi_n^{(s)}) =$$

$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + \frac{1}{q^s} \sum_{\chi_0,...,\chi_n} \chi_0(a_o^{-1})^s \cdots \chi_n(a_n^{-1})^sg(\chi_0) \cdots g(\chi_n)$$

That was my first question. And here comes my second question:

At the end they use Proposition 11.1.1 to get:

$$Z_f(u) = \frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)}$$ Here I don't see where the $$(-1)^n$$ came from.

I'm aware that you need context to answer my questions. So here are the pages:

And here is Proposition 11.1.1:

If you need something more, let me know. Thank you for your help.

• $n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$. – reuns Nov 25 '18 at 1:17