Timeline for What is the image of $-1$ by the local reciprocity map?

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Dec 1, 2018 at 5:26	comment	added	reuns		For $ [K:\mathbb{Q}_p]= 2,Gal(K/\mathbb{Q}_p) = \{ \sigma^2,\sigma\}$, or $p \equiv 1 \bmod 4 \land -1 = N_{K/\mathbb{Q}_p}(\sqrt{-1}) \land (\frac{-1}{K/\mathbb{Q}_p}) = \sigma^2 $ or $p \equiv 3 \bmod 4 \land K=\mathbb{Q}_p(\sqrt{-1}) \land -1 = N_{K/\mathbb{Q}_p}(\zeta_{2 (p+1)}) \land (\frac{-1}{K/\mathbb{Q}_p}) = \sigma^2$ or $ -1 \not \in N_{K/\mathbb{Q}_p}(K^*) \land (\frac{-1}{K/\mathbb{Q}_p}) = \sigma $
Dec 1, 2018 at 2:39	comment	added	quas93		Well, I would like for example to see how this element acts on quadratic extensions of $\mathbb{Q}_p$. If $p$ is odd, there are three of them: the unramified one, on which it acts as the identity, $\mathbb{Q}_p(\sqrt{p})$ and $\mathbb{Q}_p(\sqrt{cp})$ where $c$ is not quadratic residue. Does it act as identity on the first and minus identity on the second or is it more complicated?
Dec 1, 2018 at 2:24	comment	added	LSpice		What would count as an answer to this? (I mean, what sort of description of elements of $W$, or of elements of $W^{\mathrm{ab}}$, is more familiar?)
Dec 1, 2018 at 2:20	review	First posts
Dec 1, 2018 at 3:12
Dec 1, 2018 at 2:18	history	asked	quas93	CC BY-SA 4.0