Definition 13.10 ($s$-th unit group). For $s\in \mathbb{Z}$, the $s$-th unit group $U_K^{(s)}$ is defined by

$$U_K^{(s)}=(1+{\pi }^s{O}_K,\times ).$$

Set $U_K^{(0)}=U_K$. Then we have

$$\cdots \subseteq U_K^{(s)}\subseteq U_K^{(s-1)}\subseteq \cdots \subseteq U_K^{(0)}=U_K.$$