How does one compute the ideal class group for $ \mathbb{Z}[\sqrt{2}]$?
Motivation: I wish to prove that $ \mathbb{Z}[\sqrt{2}]$ is a PID. I have seen proofs which use norm and go on to show that it is a Euclidean domain. However, a dedekind domain is a PID if and only if it's ideal class group is trivial. I wish to use the latter method, but I'm stuck when it comes to computing the ideal class group.
Computing the ideal class group in this case is very easy. As $2=2 \mod 4$ you have that the field's discriminant is equal to $4*2=8$. Now you can compute the Minkowski bound which is in this case equal to $\frac{1}{2}\sqrt{8}$ as $r_2$ is equal to zero in a real quadratic number field. This gives you a Minkowski bound of $1,...$.
The ideal class group is generated by ideals of norm smaller than the Minkowski bound and hence is generated by only (1). Therefore we got the trivial class group.