I am going through Neukirch's algebraic number theory, when I stumbled upon this result, which is not too clear to me.
Here take $L| K$ to be a finite abelian extension of local fields, $\pi$ to be prime elements, $U^{(n)}$ the higher unit groups, and $N_{L|K}: L^{*} \mapsto K^{*}$ the norm map.
In particular, this is used in the proof of the local version of the Kronecker Weber Theorem. He argues that $(p^{f}) \times U_{\mathbb{Q}_{p}}^{(n)} \subseteq N_{L|K}(L^{*})$ for some carefully chosen $f$ and $n$. I guess more generally for a field extension $L|K$, we have $\pi_{K}^{f} \times U_{K}^{(n)} \subseteq N_{L|K}(L^{*}). $ I guess this follows since $N(L^{*})$ will contain some power of $\pi_{K}$ as $N(L^{*})$ is open of finite index, and $U_{K}^{(n)}$ form a basis of neighbourhoods of 1. Moreover, $(\pi_{K})^f \subseteq N_{L|K}(L^{*})$, since it has finite index in $K^{*}$ by the isomorphism $\text{G}(L|K) \cong K^{*}/N_{L|K}L^{*}$, so we have $\pi_{K}^{(f)} \subseteq N_{L|K}(L^{*})$.
Overall, $\pi_{K}^{(f)} \times U_{K}^{(n)} \subseteq N_{L|K}(L^{*})$.
So far so good.
He then argues that $(p^{f}) \times U_{\mathbb{Q}_{p}}^{(n)} =\bigg((p)\times U_{\mathbb{Q}_{p}}^{(n)}\bigg) \cap \bigg((p^{f})\times U_{\mathbb{Q}_{p}}\bigg)$, and he claims that the the class field corresponding to $((p^{f})\times U_{\mathbb{Q}_{p}}) $ is the unramified extension of degree $f$. I guess more generally the unramified extension of degree $n$ will have norm group equal to $(\pi_{K}^{f})\times U_{K}$, but this step is not really clear to me.
Any hints or explanations are welcome.