Norm subgroup of a 2-power cyclotomic extension of the 2-adic numbers ($N_{{\mathbb{Q}}_2({\zeta }_{2^n})/{\mathbb{Q}}_2}({\mathbb{Q}}_2({\zeta }_{2^n}{)}^{\times })$)

For every positive integer $n$,Under Local Artin reciprocity, the cyclotomic character identifies the quotient with $(\mathbb{Z}/2^n\mathbb{Z}{)}^{\times }$: a uniformizer acts trivially on the totally ramified extension, while a 2-adic unit acts through its residue class modulo $2^n$.