I have question about the exercise in Neukirch book. I think i can prove that if there exist one unit number in $\mathbb{Z}[\sqrt{2}]$ we can product infinite unit number in this way; If $m + n \sqrt{2}$ is unit then we have $m^2 - 2n^2 = 1 $ then by feremat theorem in case $n=2$ we have this another unit element $ (m^2 + 2n^2) + (2mn)\sqrt{2} $ because we have
$$ (m^2 + 2n^2)^2 + 2(2mn)^2 = (m^2 - 2n^2)^2 = 1 $$
by this proces we have that $ 3 + 2\sqrt{2} $ is unit $ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 1 $ Is this deduction wrong? Or this forms of units are in form $\pm (1+\sqrt{2})^n$ ?
