TITLE: A specific problem on Class field theory
QUESTION [6 upvotes]: Let $K$ be a quadratic complex number field. Let $p$ be a prime greater than $5$ unramified in $K/\mathbb{Q}$. Let $M$ be the compositum of all finite $p$- extensions of $K$ which are unramified outside the set of primes of $K$ lying above $p$. Let $M^{ab}$ be the maximal abelian extension of $K$ contained in $M$. Let $\Gamma^{ab}$ be Gal$(M^{ab}/K)$. For a prime $\mathfrak{P}$ of $K$ lying above $p$ let $U_{\mathfrak{P}}$ denote the local units of $K_{\mathfrak{P}}$, which is the completion of $K$ at prime $\mathfrak{P}$. Let $U_{1,\mathfrak{P}}$ denote the units that are congruent to $1$ mod $\mathfrak{P}$. Let $U_1=\prod_{\mathfrak{P}|p}U_{1,\mathfrak{P}}$. Let $H$ be the $p-$ Hilbert class field of $K$.
I want to show that (using class field theory)
we have the following exact sequence 
$1\rightarrow U_1 \xrightarrow{\beta} \Gamma^{ab} \xrightarrow{\alpha} Gal(H/K) \rightarrow 1$
It will be helpful if someone gives me a proof or a reference for this fact. The map $\alpha$ is obvious and it is also surjective but I do not know how to construct the map $\beta$ using class field theory and show that $\beta$ is injective and the sequence is exact. Thank you for help.

REPLY [3 votes]: For any number field K, CFT gives an exact sequence (with your notations; $E$ is the group of global units): 
$$E \otimes \mathbf Z_p \rightarrow U^1 \rightarrow \Gamma^{ab}  \rightarrow Gal(H/K) \rightarrow 1.$$
This is shown in an idelic way in Washington's "Introduction to cyclotomic fields", capter 13, §13-1. The image of the leftmost map is the closure of $E$, embedded in $U^1$ diagonally. Leopoldt's conjecture asserts the injectivity of this map. It is proved for abelian $K$.  For a complex quadratic $K$, $E$ is $C_2, C_4$  or  $C_6$, hence is killed by tensorization by $\mathbf Z_p$ when $p > 5$ .