An exact sequence $1 \to {O_{K,S}}^{\times}/{{O_{K,S}}^{\times}}^2 \to K(S,2) \to Cl(K,S)[2] \to 1$

Question

Let $K$ be an imaginary number field.Let $S$ be finite set of places of $K$. Let $K(S,2)\stackrel{\mathrm{def}}{=} \{b\in K^{\times}/{K^{\times}}^2 \mid v(b)≡0\mod2, \forall v\notin S \}$

Let $S-$ unit be $O_{K,S} \stackrel{\mathrm{def}}{=}\{a\in K \mid v(a)\ge 0, \forall v\notin S\}$. Let $Cl(K,S)$ be S- ideal class group of $O_{K,S}$.

I heard there is an exact sequence $1 \to {O_{K,S}}^{\times}/{{O_{K,S}}^{\times}}^2 \to K(S,2) \to Cl(K,S)[2] \to 1$. (https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/selmer_group.html)

The map ${O_{K,S}}^{\times}/{{O_{K,S}}^{\times}}^2 \to K(S,2)$ is given by $u \mapsto u mod {K^{\times}}^2$.

But what is the map $K(S,2) \to Cl(K,S)[2]$ ?

I'm sticking with this map, so I cannot prove the exactness at $K(S,2)$.

Answer 1

Hints:

1. (the sophisticated one): apply the snake lemma to the endomorphism of the exact sequence $$ 0 \rightarrow K^{\times}/O_{K,S}^{\times} \rightarrow \bigoplus_{v \notin S}{\mathbb{Z}} \rightarrow Cl(K,S) \rightarrow 0$$

given by multiplication by $2$.

2. (the unsophisticated, but also less specific one): there’s really only one natural map that makes sense. Can you guess what it is? (You take an ideal class whose square is principal, and then you want to end up with something in $K^{\times}/K^{\times 2}$… what could you do?)