The narrow class group is the group of fractional ideal modulo totally positive principal ideals. In the case of quadratic fields, this is to say positive norms principal ideals.
Consider $\sqrt{7}$, which has class number one and narrow class number two. I would like to understand how to determine explicitly éléments of each class. Aren't they exactly the positive (resp. negative) fractional ideals?
I guess not: I am reading an example where bases of these ideals (see as Z-modules) are given in the form (1,w). For the principal class, they give $$w = \frac{3+\sqrt{7}}{2}$$ and for the other class (shouldn't it be the class of the different $\sqrt{28}$?) they give $$w = \frac{5+\sqrt{7}}{3}$$ I do not understand both why it is true and how to come up with such elements.
$\Bbb{Z}+w\Bbb{Z}$ is not the same as the fractional ideal $w O_K$.
$3+\sqrt{7}$ has norm $2$ so $$\Bbb{Z}+\frac{3+\sqrt{7}}2\Bbb{Z} = \frac12(3+\sqrt{7}) O_K$$ which is a totally positive principal fractional ideal.
On the other hand $$\Bbb{Z}+\frac{5+\sqrt{7}}3\Bbb{Z}=\frac13(3\Bbb{Z}+(2+\sqrt{7}))\Bbb{Z} = \frac13 (2+\sqrt7)O_K$$ is not a totally positive principal fractional ideal.