For a tamely totally ramified extension of degree $q-1$ of a local field with residue field $\mathbb F_q$, the unit norms are exactly the principal units. Reduction sends a unit norm to its $(q-1)$st power, and the <norm-unit index in an abelian local extension> shows that this containment has the correct index.


Norm units in a tamely totally ramified extension

If $L/K$ is a finite abelian extension of local fields, then
$$
[\mathcal O_K^\times:N_{L/K}(\mathcal O_L^\times)]=e(L/K).
$$
Indeed, <Local Artin reciprocity> gives total norm index $[L:K]=ef$, while the valuation image of the norm subgroup is $f\mathbb Z$.


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 Incoming links: Norm-unit index in an abelian local extension