Cyclic local norm index ($[K^{\times }:N_{L/K}{L}^{\times }]=[L:K]$)

For a cyclic extension of p-adic fields, the Herbrand quotient of the local multiplicative group together with Hilbert theorem 90 gives this index. If the ramification index and residue degree are $e,f$, the norm valuation formula $v_K(Nx)=f{v}_L(x)$ then gives the unit norm index $e$. In particular all units are norms in an unramified extension.