Prime-to-residue-characteristic power map on principal units
= Prime-to-residue-characteristic power map on principal units
{title2=$u\mapsto u^m$}
In a complete <discrete valuation ring> of residue characteristic $p$, the power map $u\mapsto u^m$ is a bijection on $1+\mathfrak m$ when $p\nmid m$. For a <principal unit> $a$, the <Hensel lemma> lifts the simple residue root one of $T^m-a$ uniquely. This can absorb principal-unit factors in tame radical descriptions of <local fields>.