 Unit norm detects units in a local extension

For a finite extension of local fields with normalized discrete valuations, the field norm satisfies the displayed equality. The extended absolute values on a field of all conjugates agree, so $|N(x){|}_K=|x{|}_L^{[L:K]}$. Using $[L:K]=e(L/K)f(L/K)$ and $v_L{|}_K=e(L/K)v_K$ gives the equality. Thus the field norm is a unit precisely when the original element is a unit.