Serge Lang never explains anything On page 149 of Algebraic Number Theory by Serge Lang, I'm trying to understand why the inclusion $$k^{\ast}N_k^K(J_K) \cap J_c \subseteq \psi^{-1}(P_c \mathfrak N(c))$$ is true.  I've been trying for like 3 hours.  Can anyone please explain?
EDIT: What all that means: $K/k$ is an abelian extension of number fields. $c = (m(v))$ is an admissible cycle of $k$ divisible by all ramified primes, $J_c$is the subgroup of the ideles of $k$: $$\prod\limits_{v \mid c} W_c(v) \prod\limits_{v \not\mid c}' k_v^{\ast}$$ where $W_c(v) = 1 + \mathfrak p_v^{m(v)}$ for $v$ finite, and $W_c(v)$ is $(0, \infty)$ when $v$ is real.  Since $c$ is admissible, that means that $W_c(v)$ is contained in the group of local norms for all $v \mid c$.
$\phi$ is a homomorphism from $J_c$ to the group of fractional ideals which are relatively prime to $c$, given by $x \mapsto \prod\limits_{v \nmid c, v < \infty} \mathfrak p_v^{\nu_v(x_v)}$.  Also $P_c$ is the group of principal ideals $x \mathcal O_k$, where $x \in J_c \cap k^{\ast}$ (in more standard notation, $x \equiv 1 \mod^{\ast} c$).  Finally $\mathfrak N(c)$ is the group of norms of fractional ideals of $K$ which are relatively prime to $c$.
 A: $\newcommand{\fraka}{\mathfrak{a}} \newcommand{\frakA}{\mathfrak{A}} \newcommand{\frakc}{\mathfrak{c}} \newcommand{\frakN}{\mathfrak{N}} \newcommand{\frakb}{\mathfrak{b}}$
Let $\alpha \in J_\mathfrak{c}$ be an idèle that's relatively prime to $\mathfrak{c}$, such that $\alpha = aN_{K/k}(\beta)$ for some $\beta\in J_K$ and $a\in k^*$. We want to show $\psi(\alpha)\in P_\frakc \frakN_\frakc$.
We have $\psi(a)\in P$, and $\psi(N_{K/k}(\beta))\in \frakN$, so $\psi(\alpha) \in P\frakN$. Furthermore, since $\alpha\in J_\frakc$, we also have $\psi(\alpha) \in I(\frakc)$. It's then enough to show that $I(\frakc)\cap P\frakN \subset P_\frakc \frakN_\frakc$.
So let $\frakb\in \frakN$ and $(a)\in P$ be such that $(a)\frak{b}$ is relatively prime to $\frakc$ (e.g. $\frak{b} = \psi(\beta)$ and $(a)=\psi(a)$ in the above.)
Now by the approximation theorem, there exists some $\gamma\in K^\times$ such that $N_{K/k}(\gamma)\frakb$ is prime to $\frakc$. For instance for all $v|\frakc$, $v<\infty$, if $w_1,\cdots,w_r$ are the places over $v$, we can pick $\gamma$ such that $w_1(\gamma)=-v(\frakb)$ and $w_i(\gamma)=0$ for $i>1$. Now let $a' = aN_{K/k}(\gamma^{-1})$ and $\frakb' = N_{K/k}(\gamma)\frakb$, so that $a'\frakb'=a\frakb$. Then $\frakb'$ and $a'\frakb'$ are relatively prime to $\frakc$, so $(a')$ is also relatively prime to $\frakc$. In other words $(a')\in P_\frakc$, $\frakb'\in \frakN_\frakc$ and so $a\frakb=a'\frakb'\in P_\frakc\frakN_\frakc$.
