I have a seemingly easy question concerning the Artin Reciprocity as stated in the book Advanced Topics in the Arithmetic of Elliptic Curves by Silverman. He states it as follows.
Let $L/K$ be a finite abelian extension of number fields. There exists an integral ideal $\mathfrak{c} \subseteq \mathcal{O}_K$, divisible by precisely the primes of $K$ that ramify in $L$, such that \begin{align*} (\alpha \mathcal{O}_K , L/K) = 1 \;\;\; \textit{for all $\alpha \in K^{\times}$ satisfying $\alpha \equiv 1 \;(mod\; \mathfrak{c})$}. \end{align*} Here $(\; \cdot \;, L/K)$ denotes the Artin map $I_K(\mathfrak{c}) \longrightarrow \text{Gal}(L/K)$, where $I_K(\mathfrak{c})$ is the group of fractional ideals of $\mathcal{O}_K$ which are relatively prime to $\mathfrak{c}$. First of all, I am unsure about what the congruence means, since $\alpha$ does not necessarily belong to $\mathcal{O}_K$. Moreover, I wonder why it makes sense to evaluate the Artin map at $\alpha \mathcal{O}_K$, i.e. why none of the primes in the factorization of $\alpha \mathcal{O}_K$ appears in the one of $\mathfrak{c}$...
Thanks in advance for any comment.
UPDATE: By consulting Algebraic Number Theory by Lang, Chapter VI, the question on the special congruence got solved and with it the problem on evaluating the Artin map.