TITLE: Proof of a theorem in Washington's book on Cyclotomic fields.
QUESTION [1 upvotes]: I am reading chapter 13 - 'Iwasawa theory of $\mathbb Z _p$-extensions' in Washington's book 'Cyclotomic fields'.
On page 267 it is claimed that for extension $F$ (which is infinite as it is maximal unramified extension outside $p$), there is a closed subgroup $H \subset J$(the idèle group) containing $K^*$ such that $J/H \simeq Gal(F/K)$.
Milne states the surjectivity of the reciprocity map for number fields (once the map is subjective the existence of closed subgroup is fine as the reciprocity map is continuous and the subgroup $\{1\}$ is closed in Krull topology). So I am ok with this.
Later on the same page the author claims that for $$U''=\prod _{\mathfrak p \nmid p} U_{K_\mathfrak p } $$ $\overline {K^* U''} \subset H$ and because $F$ is maximal, and this is the part which I do not follow, $H=\overline {K^* U''}$.
Can I prove that under the reciprocity map if $I_K/\overline {K^*U''} \simeq Gal(L/K)$ then $L$is unramified outside $p$ ?
Any help is appreciated.

REPLY [1 votes]: In Milne's notes, V, 5.7, it is explained what happens for infinite extensions. In the number field case, the map from the idele group is always surjective with, as you observe, closed kernel. The kernel is always the closure of the obvious stuff that it contains. I don't think that's proved in Milne's notes --- perhaps in Tate's article in Cassels-Frohlich or Artin-Tate. (Obvious stuff: $K^{\times}$ times the product of whatever is in the kernels of the local Artin maps.)
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