When cocycle $G_K\to \mu_2, \sigma \mapsto \alpha^{\sigma}/\alpha$ is unramified at $v$?

Question

Let $K$ be a number field. Let $G_K$ be absolute Galois group of $K$. Let $M_K$ be a set of all places of $K$. Fix an element $a\in K^{\times}/{K^{\times}}^2$.

When cocycle $f: G_K\to \mu_2, \sigma \mapsto \alpha^{\sigma}/\alpha$ ($\alpha \in \overline{K}^{\times}$ such that $\alpha^2=a)$ is unratified at $v$ ?

Here, we say cocycle is unramified if only if $f\mid_{I_v}$ is trivial where $I_v$ is inertia group.

Silverman's book 'The arithmetic of elliptic curves' essentially says this is when $v(\alpha)≡0\text{mod}2$.

But how can we prove $f$ is unramified if only if $v(\alpha)≡0\text{mod}2$ ?

Answer 1

That criterion only applies to places $v\nmid 2\infty$.

The cocycle factors through the Galois group of $K(\sqrt{a})/K$, i.e. it is trivial on the absolute Galois group of $K(\sqrt{a})$. For $\sigma$ in the Galois group of $K(\sqrt{a})/K$, the cocycle is trivial on $\sigma$ if and only of $\sigma(\sqrt{a})=\sqrt a$, i.e. if and only if $\sigma$ is trivial. The same analysis also applies locally. So the cocycle is ramified at $v$ if and only if the unique non-trivial element of the Galois group of $K(\sqrt{a})/K$ is in the inertia group at $v$, i.e. if and only if the extension $K(\sqrt{a})/K$ is ramified at $v$, which at finite places not above $2$ is equivalent to the stated criterion.

Answer 2

By Hilbert's 90 the short exact sequence $$1\to\mu_2\to\overline K_v^\times\xrightarrow2\overline K_v^\times\to 1,$$ gives rise to long exact sequences $$1\to \mu_2\to K_v^\times\xrightarrow2 K_v^\times\to H^1(G_{K_v},\mu_2)\to 1$$ and $$1\to \mu_2\to (K_v^{unr})^\times\xrightarrow2 (K_v^{unr})^\times\to H^1(I_{K_v},\mu_2)\to 1.$$ Thus, there are isomorphisms $$H^1(G_{K_v},\mu_2)\simeq K_v^\times/(K_v^\times)^2,\hspace{0.3cm}H^1(I_{K_v},\mu_2)\simeq (K_v^{unr})^\times/((K_v^{unr})^\times)^2.$$ In fact, there is even an isomorphism $$ v\colon (K_v^{unr})^\times/((K_v^{unr})^\times)^2\xrightarrow\sim\mathbb Z/2.$$ There is a restriction map $H^1(G_{K_v},\mu_2)\to H^1(I_{K_v},\mu_2)$, and a cocyle is unramified iff its image in $H^1(I_{K_v},\mu_2)$ is trivial. In other words, a $\alpha\in K_v^\times$ gives rise to an unramified cocycle iff $\alpha\in ((K_v^{unr})^\times)^2$, i.e., $v(\alpha)\in2\mathbb Z$.