I want to know whether for any global number field $k$, the closed subgroup of the idele corresponding to the maximal elementary p-extension(p is a prime number) is k*$J^p$.

 The critical point is to show that the subgroup k*$J^p$ is closed.

 If so, as the idele group is locally compact, the quotient group is totally disconnected such that k*$J^p$ is the intersection of open subgroups of $J$ containing k*$J^p$. Using the compactness of the quotient group, we see that k*$J^p$ corresponds to an infinite abelian extension. By the group theory, the abelian extension must be the maximal elementary p-extension.

I checked this to be true when $k$ is $Q$ or an imaginary quadratic extension. In these cases the above is true because the unit group is finite. For general number fields I haven't checked this yet.

So I hope someone can give me a reference or show me a proof or a counterexample.

Thank you