Unit norm detects units in a local extension
= Unit norm detects units in a local extension
{title2=$v_K(N_{L/K}x)=f(L/K)v_L(x)$}
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.