Dedekind zeta function for a subfield of cyclotomic field

Can someone give me a proof or a reference of a proof of Proposition 23 here. I cite it below. $\mu_n$ is an $n$th root of unity.

Proposition 23. Let $F$ be a number field contained in $\mathbb{Q}\left(\mu_{n}\right)$ for some $n \in \mathbb{Z}^{+} .$Identify $G_{n}=\operatorname{Gal}\left(\mathbb{Q}\left(\mu_{n}\right) / \mathbb{Q}\right)$ with $(\mathbb{Z} / n \mathbb{Z})^{\times}$and let $X_{F} \leq G_{n}$ be the subgroup of Dirichlet characters whose kernel contains the subgroup of $G_{n}$ fixing $F$. Then $$ \zeta_{F}(s)=\prod_{\chi \in X_{F}} L(s, \chi) $$ where the $L$-function $L(s, \chi)$ is defined to be $$ L(s, \chi)=\sum_{n=0}^{\infty} \frac{\chi(n)}{n^{s}}=\prod_{q}\left(1-\frac{\chi(q)}{q^{s}}\right)^{-1} $$

I don't know where to start with the proof, I had only seen this for the cyclotomic field itself.

Solution 1:

Here is the standard story:

Notice that $X_F$ is the group of characters on the abelian group $G_F:=Gal(F/\mathbb Q)$, and that we have a surjective homomorphism $G_n=Gal(\mathbb Q(\mu_n)/\mathbb Q)=(\mathbb Z/n\mathbb Z)^\times\to G_F$. If $p$ doesn't divide $n$ (i.e. is unramified in $\mathbb Q(\mu_n)$) then $[p]\in G_n$, and thus also its image in $G_F$, is the Frobenius element $Frob_p$.

If $p$ is an unramified prime then the local factor $\prod_{\chi\in X_F}(1-\chi(p=Frob_p)\,p^{-s})^{-1}$ in your formula for $\zeta_F(s)$ can be shown to be equal to $(1-p^{-f_ps})^{-g_p}$, where $f_p$ is the order of $p=Frob_p$ in $G_F$ and $g_p=|G_F|/f_p$, i.e. to the usual local factor of $\zeta_F$.

A way of showing that equality $$\prod_{\chi\in X_F}(1-\chi(p)\,p^{-s})^{-1}=(1-p^{-f_ps})^{-g_p}:\qquad(*)$$ Both sides can be seen as a determinant of a suitable $T_p:W\to W$, where $W$ is the space of complex functions on $G_F$. Namely, we have the (regular) action $\rho$ of $G_F$ on $W$ given by ($\gamma,\epsilon\in G_F$, $r\in W$) $(\rho(\gamma)\, r)(\epsilon)=r(\gamma\epsilon)$, and $T_p=(1-\rho(p)\,p^{-s})^{-1}$. Indeed, in the basis $X_F$ of $W$ the endomorphism $T_p$ is a diagonal matrix, and the LHS of $(*)$ is the product of its diagonal elements. On the other hand, seeing $T_p$ in the basis of $\delta$-functions gives "easily" the RHS (split $G_F$ to the $g_p$ orbits of the action of $p$, then compute the determinant for one orbit).

I skipped what happens at the ramified primes - the story is basically the same, but $W$ needs to be replaced by its $I_p$-invariant part, where $I_p$ is the inertia group.