I have a question about the proof of Corollary (1.9), Chap V page 324 from Jürgen Neukirch's Algebraic Number Theory:
Claim: Every finite abelian extension of $L \vert \mathbb{Q}_p$ is contained in a field $\mathbb{Q}_p(\zeta)$, where $\zeta$ is a root of unity. In other words: The maximal abelian extension $\mathbb{Q}_p^{ab} \vert \mathbb{Q}_p$ is generated by adjointed roots of unity.
Proof: For suitable $f$ and $n \in \mathbb{N}$, we have $(p^f) \times U^{(n)}_{\mathbb{Q}_p} \subset N_{L \vert \mathbb{Q}_p} L^*$. Therefore $L$ is contained in the class field / [the proof continues...]
About used notations: $(p) \subset \mathbb{Z}_p$ is the unique maximal ideal of $p$-adic integers and $(p^f)$ it's $f$-th power. $U^{(n)} = 1 +(p)^n$ and $N_{L \vert \mathbb{Q}_p} L^* \subset \mathbb{Q}_p$ is the image of $L^*$ under norm map.
Question: why such $f, n$ exist and now they can be found or their existence be showed?
Here $(p)$ is not $p \Bbb{Z}_p$ but $p^\Bbb{Z}$.
With $f = [L:\Bbb{Q}_p]$ and $p^{n-2}$ the largest power of $p$ dividing $f$ $$p^f= N_{L/\Bbb{Q}_p}(p), \qquad 1+p^n \Bbb{Z}_p= (1+p^2 \Bbb{Z}_p)^{p^{n-2}}=(1+p^2 \Bbb{Z}_p)^f=N_{L/\Bbb{Q}_p}(1+p^2\Bbb{Z}_p)$$