Let $K/k$ be an abelian extension of number fields and $\frak c$ a cycle of $k$ divisible by all primes ramifying in $K$. Then we have the following theorem from algebraic number theory (see Lang's ANT chapter X, Theorem 1):
The Artin map $A : I({\frak c})\to \operatorname{Gal}(K/k),{\frak a}\mapsto ({\frak a},K/k)$ is surjective.
The proof in Lang's book goes like this:
Consider the fixed field $F$ of the image of $A$. Then every prime in $I(c)$ splits completely in $K$. If $A$ is not surjective, i.e. $F\ne k$ there is an intermediate field $F_0$ of $F/k$ which is cyclic and of degree $>1$ over $k$. He now refers to the global cyclic norm index equality and one of its corollaries (from chapter IX) to deduce that in such an extension infinitely many primes do not split completely which is a contradiction because $I(\frak c)$ contains all but finitely many primes from $k$.
The global cyclic norm index equality was a very non-trivial theorem the proof of which was basically the whole chapter IX.
It seems like we only need the following:
If $L/k$ is finite and Galois then the set of primes of $k$ splitting completely in $L$ has density $\frac{1}{[L:k]}$. Thus (in the setting above) we have $[F:k]=1$.
This can be deduced very easily from the more elementary fact that the Dedekind zeta function of $L$ has a simple pole at $1$ (see below). And this holds for arbitrary (finite), not necessarily abelian or even cyclic, Galois extensions.
This also shows that in general, i.e. if $K/k$ might not be abelian, but still finite and Galois, the set of Frobenius elements of primes in $I(\frak c)$ generates the whole group $\operatorname{Gal}(K/k)$ (of course this can be improved by replacing 'generates' by 'equals' using e.g. Chebotarev's density theorem which is again more involved)
Why did Lang use the more complicated/involved proof?
I know that he already used some class field theoretical facts in earlier results on prime densities (e.g. in the theorem on primes in generalized arithmetic progressions or the Chebotarev density theorem) but this is not the case here, so no circular reasoning.
Are there maybe other situations (e.g. other global fields, i.e. function fields over $\Bbb F_q$) where Lang's argument is advantageous?
Proof of the claim above:
We have \begin{align*}
\log(s-1)^{-1}&\sim\log\zeta_L(s)\\
&\sim\sum_{{\frak P}\subset L}N{\frak P}^{-s}\\
&\sim\sum_{\substack{{\frak P}\subset L\\f({\frak P}\mid{\frak P}\cap k)=1}\\e({\frak P}\mid{\frak P}\cap k)=1}N{\frak P}^{-s}\\
&=\sum_{\substack{{\frak P}\subset L\\{\frak P}\cap k\text{ splits completely in $L$}}}N{\frak P}^{-s}\\
&=[L:k]\sum_{\substack{{\frak p}\subset k\\{\frak p}\text{ splits completely in $L$}}}N{\frak P}^{-s}\\
\end{align*}
It follows that $\{{\frak p}\subset k\mid {\frak p}\text{ splits completely in }L\}$ has (Dirichlet-)density $\frac{1}{[L:k]}$.
(Here $\sim$ denotes that the difference of the two sides is bounded as $s\to 1^+$.)