Serge Lang Never Explains Anything Round II

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

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).

Attribution
Source : Link , Question Author : Bless You Two , Answer Author : Zavosh

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