 Successive quotients of principal-unit groups

For $r\ge 1$, the map $1+{\pi }^{r}a\mapsto \bar{a}$ is a surjective group homomorphism to the additive residue field with kernel $U_{r+1}$. The extra product term is divisible by ${\pi }^{r+1}$. Thus each quotient has cardinality $|k|$, a useful count when splitting the unit group by roots of unity.