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I have been reading into class field theory lately and I would say, that I'm understanding, how the proofs for most of the theorems work, but I'm struggling a lot with the intuition.

One of my big problems is the Albert-Brauer-Hasse-Noether theorem and the exact sequence $$ 0\to \mathrm{Br}(L/K)=H^2(G,L^*)\to \bigoplus_{v}\mathrm{Br}(L^v/K_v)\overset{\sum \mathrm{inv}}{\to}\frac{1}{n}\mathbb{Z}/\mathbb{Z}(\to 0)$$ for a Galois extension $L/K$ of global fields of degree $n$. (I denote by $L^v$ the completion of $L$ at any place of $L$ lying above $v$ as Milne does in his lecture notes.) Assume that $L/K$ is cyclic, then $H^2(G,L^*)=K^*/N_{L/K}(L^*)$ and $\mathrm{Br}(L^v/K_v)=K_v^*/N_{L^v/K_v}((L^v)^*)$. For the unramified places, the map $\mathrm{inv}$ is then given by the valuation. I have two questions:

  1. Is there a good intuition for why an element $x\in K^*$, the sum $\sum_v \mathrm{inv}(x_v)$ is $0$? (I know the proof (for example in the book by Milne or Neukirch) reducing to cyclotomic extensions of $\mathbb{Q}$ and then proving it by explicit calculations.)
  2. Given a sequence $a_v/n_v$ with sum $0$ and only finitely many terms non-zero, how do you construct an element $x\in L^*$, which induces exactly this sequence in $\bigoplus_{v}\mathrm{Br}(L^v/K_v)$? (For simplicity on can assume that is supported only on unramified places.)

Or do you know of some references, that go into these types of questions?

Thank you very much for your help in advance!

Firebolt2222
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Dec 16 '24 at 11:35
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    I think if you take $n=2$ and $K=\mathbb Q$, the resulting short exact sequence is more or less equivalent to the law of quadratic reciprocity. Generally, the fact that the residues sum to zero is often regarded as a form of "higher reciprocity law". At least this is what I recall reading in a short expose by David Harari, which unfortunately I can't find right now (but might be worth to keep in the back of your mind, he is a very good and instructive writer). – R.P. Dec 16 '24 at 12:56
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    Regarding the text by David Harari, I think it was this one, unfortunately it is in French: https://www.imo.universite-paris-saclay.fr/~david.harari/articles/newgazette.pdf But of course you could run it through a translator app. – R.P. Dec 16 '24 at 13:13
  • Thank you @R.P.! I will have a look at the notes. My French is not so good, but I'll give it a shot. – Firebolt2222 Dec 16 '24 at 13:32
  • I looked it up myself in the book "Galois Cohomology and Class Field Theory" by David Harari and your comment about quadratic reciprocity is spelled out nicely in section 14.2. – Firebolt2222 Dec 16 '24 at 14:34

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