Image of the idele class group and its subgroup of idelic norm 1

Question

[Sorry if the title isn't specific, it was too long.]

My question is: Why does $J_{K}/J_{K}^{1}\cong %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ imply that $J_{K}/K^{\ast }$ and $J_{K}^{1}/K^{\ast }$ have the same image under any continuous homomorphism from $J_{K}$ to a discrete group $G$?

Here $J_{K}$ denotes the group of ideles on a number field $K$ and $% J_{K}^{1}$ the subgroup of elements with adelic norm $1$. I know it has something to do with the fact that $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is connected, but don't know how to proceed from there.

Thank you.

Answer 1

If $\phi: H \to G$ is a map from a topological group to a discrete topological group, then $\phi$ kills the connected component of the identity in $H$ (proof: the inverse image of the identity of $G$ is closed and open and contains the identity, hence contains the connected component of the identity).

But anything in $J_K$ can be moved to something in $J_K^1$ by multiplying by something in the connected component of the identity. (e.g. by modifying just one archimedean place.)