Units congruent to one modulo 3 plus square root two
= Units congruent to one modulo 3 plus square root two
{title2=$\{u\in\mathbb Z[\sqrt2]^\times:u\equiv1\pmod{3+\sqrt2}\}$}
Reduction modulo $(3+\sqrt2)$ identifies $\mathbb Z[\sqrt2]/(3+\sqrt2)$ with $\mathbb F_7$, sends $\sqrt2$ to $4$, and sends $\varepsilon=1+\sqrt2$ 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
$$
\left\langle-(1+\sqrt2)^3\right\rangle
=\langle-7-5\sqrt2\rangle.
$$