<Local Artin reciprocity> states that the continuous map
$$
\operatorname{Art}_K:K^\times\longrightarrow\operatorname{Gal}(K^{\mathrm{ab}}/K)
$$
has dense image and induces, for every finite abelian extension $L/K$, an isomorphism
$$
K^\times/N_{L/K}(L^\times)\cong\operatorname{Gal}(L/K).
$$
Thus the <norm subgroup of a local field extension> is the kernel of the Artin map restricted to $L$.

For $L=K_n$, the uniformizer $p$ maps to the unramified Frobenius and therefore acts trivially on the totally ramified cyclotomic extension. By part b, a unit $u$ acts trivially on $K_n$ exactly when $u\equiv1\pmod{p^n}$. Since
$$
\mathbb Q_p^\times=p^{\mathbb Z}\times\mathbb Z_p^\times,
$$
the kernel, and hence the norm subgroup, is
$$
N_{K_n/\mathbb Q_p}(K_n^\times)
=\langle p\rangle\times(1+p^n\mathbb Z_p).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 136 / 5 / c / Solution

{title2=$N_{L/K}(L^\times)$}

For a finite extension of local fields $L/K$, its norm subgroup is the image of the <field norm> $N_{L/K}:L^\times\to K^\times$. For an abelian extension, it is the kernel of the induced local Artin map to $\operatorname{Gal}(L/K)$.

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 Incoming links: Norm subgroup of a local field extension