Serge Lang Never Explains Anything Round II

Question

I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote:

Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian extension with Galois group $G$. Then there exists a number field $k$ and an abelian extension $K$, with absolute value $v$, such that $$ F = k_v, L = K_v$$ For instance, let $E$ be a number field dense in $L$. Let $K$ be the composite of $\sigma E$ for all $\sigma \in G$. Then $K$ is stable under $G$ and we let $k$ be the fixed field of $G$. It is immediate that $k_v = F$, and of course $K_v = L$.

Ookay, so far so good. Then he drops this gem:

Note that the local Artin map $k_v^{\ast} \rightarrow G(K k_v/k_v)$ is induced by the global map. The consistency property of the global symbol implies that the local map is independent of the global extension $K$ over $k$ chosen such that $K_v = L$ and $k_v = F$.

'Consistency' means that for a bigger abelian extension $M$ of $k$ containing $K$, that the restriction of $(x, M/k)$ to $K$ is $(x, K/k)$. But this doesn't explain at all why the local Artin map is independent of the global parameters. You would need to show that for a different abelian extension $K'/k'$ such that $K'_w = L$ and $k'_w = F$, then $(x, K'/k')$ and $(y, K/k)$ can be identified as the same element of $G$ for $x, y$ suitably identified in $k'$ and $k$. Any help here?

P.S. I actually really like Serge Lang's treatment of ANT, loved his complex analysis textbook, it's just frustrating at parts because he assumes you're a Level 99 Clever Warlord.

Answer 1

Suppose that $K/k$ and $K'/k'$ are two abelian extensions of number fields, with valuations $v$ and $v'$ such that $K_v=K'_{v'}=L$, and $k_v=k'_{v'}= F$. We want to show that for $a\in F^*$, we have $(a,K/k)=(a,K'/k')$, when $a$ is alternately viewed as a local idele in $J_k$ or in $J_{k'}$.

Since the Artin symbol is compatible with isomorphisms of fields (property $\mathbf{A1}$ in Lang, p.207), we can assume $K$ and $K'$ are both contained in a larger number field. Let $\tilde{k}=kk'$, pick $w$ a place of $\tilde{k}$ lying over $v$, and take $(\tilde{k}_w,w)$ to be the corresponding completion. Since the topologies of $(k_v,v)$, $(k'_{v'},v')$ and $(\tilde{k}_w,w)$ agree, we have $\tilde{k}_w = k_v k'_{v'} = F$.

Let $a\in F^*$, and consider $a$ as an element of $\tilde{k}_v^* \subset J_{k}$. Since $\tilde{k}_w = k_v$, we have $a = N_{\tilde{k}/k}(b)$ for some local idele $b\in \tilde{k}_w^* \subset J_{\tilde{k}}$, namely the image of $a$ under the inclusion $k_v = \tilde{k}_w \subset J_{\tilde{k}}$. Now by formal properties of the global Artin map (property $\mathbf{A3}$ of Lang, p. 208) we have $(a,K/k) = (N_{\tilde{k}/k}(b), K/k) = \mathrm{res}_K (b, K\tilde{k}/\tilde{k}).$ Since $k_v = k'_{v'}$ and $b$ is local, we also have $N_{\tilde{k}/k'}(b)=a\in k'_{v'}$ so for the same reason $(a,K'/k') = (N_{\tilde{k}/k'}(b),K'/k') = \mathrm{res}_{K'} (b,K'\tilde{k}/\tilde{k}).$ Then $(a,K/k)=(a,K'/k')$ would follow from $(b,K\tilde{k}/\tilde{k})=(b,K'\tilde{k}/\tilde{k})$. This shows that, since $K\tilde{k}/\tilde{k}$ and $K'\tilde{k}/\tilde{k}$ are abelian, to solve the problem it suffices to assume $k=k'$.

If $k=k'$, then $\tilde{K}=KK'$ is an abelian extension of $k$ containing both $K$ and $K'$. The result now follows from the consistency property of the Artin symbol ($\mathbf{A2}$ in Lang, p.208). On the one hand $(a,K/k) = \mathrm{res}_K (a,\tilde{K}/k)$, and on the other hand $(a,K'/k) = \mathrm{res}_{K'} (a,\tilde{K}/k)$. Now $a\in J_k$ is a local idele in $k_v^*=k_{v'}^*$, and $K_v = K'_{v'}$, so we have $\mathrm{res}_K (a,\tilde{K}/k)= \mathrm{res}_{K'} (a,\tilde{K}/k)$.