Successive quotients of principal-unit groups ($U_r/U_{r+1}\cong (k,+),U_r=1+{\pi }^r{\mathcal{O}}_K$)

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.