I have been looking at several different proofs that Artin $L$-series of Abelian extensions coincide with Hecke $L$-series.
In Serge Lang's $\textit{Algebraic Number Fields}$ (XII §2) and Jürgen Neukirch's $\textit{Algebraische Zahlentheorie}$ (VII §10) they use a property which I paraphrase in the following manner:
Let $E/K$ be an Abelian extension, $G:=\textrm{Gal}(E/K)$, and let $\chi$ be a simple character of $G$. We then have $G/\textrm{Ker}(\chi) \cong \textrm{Gal}(E_{\chi}/K)$, where $E_{\chi}$ is the fixed subfield corresponding to $\textrm{Ker}(\chi) \vartriangleleft G$. By inflation, we may view $\chi$ is a character of $\textrm{Gal}(E_{\chi}/K)$, and we have: $$ L(E_{\chi}/K,\chi,s) = L(E/K,\chi,s) $$
I have some scruples with this argument.
I understand the property of inflation to mean the following:
Let $E/K$ be a Galois extension, $G:=\textrm{Gal}(E/K)$, and let $E'/K$ be a bigger Galois extension ($E \subset E'$), $G':=\textrm{Gal}(E'/K)$, and let $\chi$ be a simple character of $G$. We then have: $$ L(E'/K,\chi',s) = L(E/K,\chi,s) $$ $\chi' = \chi \circ \pi$, where $\pi: G' \to G$ is the canonical projection.
That is, inflation allows us to pass $\textit{from a smaller Galois extension to a bigger one}$.
But Lang and Neukirch (et al.) seem to be going the other way: They take a character of a bigger Galois extension and pass to a smaller one.
But this is manifestly impossible. Take $\textit{e.g.}$ $\mathbb{Q} / \mathbb{Q}$ for the smaller extension and $\mathbb{Q}(i) / \mathbb{Q}$ for the bigger one. The above would imply that: $$ L(\mathbb{Q}(i) / \mathbb{Q}, \chi, s) = L( \mathbb{Q} / \mathbb{Q} , \chi , s) = \zeta(s) $$ On the left, we could let $\chi$ be the non-trivial sign character. On the right, we must necessarily have the trivial character.
How is this to be understood?
Thank you for your attention.
