= Norm subgroup of a 2-power cyclotomic extension of the 2-adic numbers
{title2=$N_{\mathbb Q_2(\zeta_{2^n})/\mathbb Q_2}(\mathbb Q_2(\zeta_{2^n})^\times)$}
For every positive integer $n$,
$$
N_{\mathbb Q_2(\zeta_{2^n})/\mathbb Q_2}
\bigl(\mathbb Q_2(\zeta_{2^n})^\times\bigr)
=2^{\mathbb Z}(1+2^n\mathbb Z_2).
$$
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 $2^n$.
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