The theory of ring class fields corresponding to orders in imaginary quadratic fields is introduced by, say, [1]. But I'm reading an article [2], in which ring class fields corresponding to orders in general number fields are used.
For example, let $d$ be a positive integer and $F=\mathbb Q(\sqrt{-d})$, $\mathfrak o_F$ be the integral ring of $F$, and $E = F(\sqrt{-1})$. Take the order $L=\mathfrak o_F+\mathfrak o_F\sqrt{-1}$ inside $E$. Let $\mathfrak p$ be a place of $F$ and $L_{\mathfrak p}$ be the $\mathfrak p$-adic completion of $L$ inside $E_\mathfrak p=E\otimes_F F_\mathfrak p$. Define the ring class field $H_L$ corresponding to $L$ to be the class field corresponding to the open subgroup $E^\times\prod_{\mathfrak p} L_\mathfrak p^\times$ of the Id`ele $\mathbb I_E$ by class field theory, s.t. $$\mathbb I_E/(E^\times\prod_{\mathfrak p} L_\mathfrak p^\times)\cong Gal(H_L/E)$$
My question is:
Let $Cl_L$ be the Picard group of $L$, then is it true that $$\mathbb I_E/(E^\times\prod_{\mathfrak p} L_\mathfrak p^\times)\cong Cl_L?$$ And why? Thanks.
[1] Cox, David A. "Primes of the form x2+ ny2, A Wiley-Interscience Publication." New York (1989).
