I have two question on adeles and ideles:
$1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$.
$2)$ Also it will be helpful if someone explains me in details the following : The topologyof $I_K$ obtained as a subspace topology of $A_K \times A_K$ is the same as that of restricted product with $A_K^{\ast}$ with $O_v^{\ast}$. Here $A_K$ is the adeles group of K
There is a surjective absolute value morphism $I_K \to \mathbb R^{\times}_{>0}$ (given by taking the product of the absolute values on each factor), which (by the product formula) has $K^{\times}$ in its kernel.
Thus $I_K/K^{\times}$ surjects onto $R^{\times}_{> 0}$, and so is not compact. On the other hand, the kernel of this surjection is compact. (This is equivalent to the finiteness of the class group together with Dirichlet's unit theorem.)
As for your second question, have you tried just writing it all out?