My question concerns the proof of Proposition 10.4. in chapter II of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves. The situation is the following.
Let $E/L$ be an elliptic curve with CM by the ring of integers $\mathcal{O}_K$ ($K$ is quadratic imaginary) and denote its associated Grössencharacter on the ideles by $\psi_{E/L}$. Given a prime $\mathfrak{P}$ of $L$ at which $E$ has good reduction, the statement is that the endomorphism on $E$ given by multiplication by $\psi_{E/L}(\mathfrak{P})$ reduces (modulo $\mathfrak{P}$) to the Frobenius $\phi_{\mathfrak{P}}$ on the reduction of $E$.
He fixes an idele $x \in \mathbb{A}_L^{\times}$ with a uniformiser in its $\mathfrak{P}$-component and $1$'s elsewhere. When considering the reduction modulo $\mathfrak{P}$, he writes We have $[x,L] = (\mathfrak{P}, L^{ab}/L)$ from (3.5), so $[x,L]$ reduces to the $N\mathfrak{P}$-power Frobenius map.
My confusion starts with the equality $[x,L] = (\mathfrak{P}, L^{ab}/L)$. First of all, he introduces this Frobenius symbol (a few sections earlier) only for finite abelian extensions. I am aware that this notion could maybe be lifted to infinite extensions by considering limits (and the resulting symbol is well-defined up to limit of the inertia subgroups I guess...). Is that what he does? Because (and this is the second point of confusion) the statement he refers to (chapter II, Theorem 3.5, part (c) I guess, page 120) is formulated for any abelian extension, with the assumption that a given prime is unramified in that extension (and then giving an equality as above). So here comes the third part of confusion: What does it mean for a prime to be unramified in an infinite abelian extension (for example $L^{ab}/L$)?
Thanks in advance for any comment.