Let us fix a square free positive integer $n\in\mathbb{N}$ and consider the number field $\mathbb{Q}(\sqrt n)$ with ring of integers $K=\mathbb{Z}[\sqrt n]$. Let us denote the Galois norm of elements in this ring by $\mathcal{N}$. Every invertible element in $K$ has to satisfy $$\mathcal{N}(a+b\sqrt n)=a^2-b^2n=1.$$ Moreover, these elements form the group of units $K^\times$, which thanks to Dirichlet's unit theorem can be expressed as $(\mathbb{Z}[\sqrt n])^\times = \{\pm1\}\times \langle\epsilon\rangle$ for a fundamental unit $\epsilon\in\mathbb Z[\sqrt n]$.
Let us now fix some prime $p$ and $k\in\mathbb N$. I wonder if one can find an analogous description of elements of norm 1 in $(K/p^kK)^\times$: more formally, the Galois norm descends mod $p^k$ to a map $\mathcal{N}_{p^k}:(K/p^kK)^\times\to (\mathbb{Z}/p^k\mathbb{Z})^*$ and we consider the group $$\mathcal{I}_{p^k}=\{a+b\sqrt n \in(K/p^kK)^\times:\mathcal{N}_{p^k}(a+b\sqrt n)=1\pmod p^k\}.$$ Can one describe it explicitly? In fact, can it be equal to $$\{1,p^k-1\}\times \langle\epsilon_{p^k}\rangle$$ for $\epsilon\equiv \epsilon_{p^k}\pmod{p^k}$?
