Units congruent to one modulo 3 plus square root two ($\{u\in \mathbb{Z}[\sqrt{2}{]}^{\times }:u\equiv 1(\mathrm{mod}3+\sqrt{2})\}$)

Reduction modulo $(3+\sqrt{2})$ identifies $\mathbb{Z}[\sqrt{2}]/(3+\sqrt{2})$ with ${\mathbb{F}}_7$, sends $\sqrt{2}$ to $4$, and sends $\epsilon =1+\sqrt{2}$ to $5$. Since $5$ has order six and $5^3=-1$ in ${\mathbb{F}}_7^{\times }$, the kernel on units is the infinite cyclic group