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

Local Artin reciprocity states that the continuous maphas dense image and induces, for every finite abelian extension $L/K$, an isomorphismThus 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 $u\equiv 1(\mathrm{mod}{p}^n)$. Sincethe kernel, and hence the norm subgroup, is