Principal-unit filtration of a local field
= Principal-unit filtration of a local field
{title2=$U_m=1+\mathfrak p^m\quad(m\ge1)$}
The groups $U_m$ form a compact-open neighbourhood basis at one inside the units of a non-Archimedean <local field>. A <smooth character> trivial on $U_{i+1}$ but nontrivial on $U_i$ cannot have a nonzero fixed vector under $U_m$ for $m\le i$. This elementary conductor comparison controls admissibility and smooth duals of direct sums.