# Understanding the narrow class group

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.
• @DesideriusSeverus Just check that it works. $O_K=\Bbb{Z}[\sqrt7]$ so $aO_K=a\Bbb{Z}+a \sqrt7 \Bbb{Z}$. Equivalenty check that it is a $O_K$-module containing $a$ and contained in $aO_K$. Nov 14, 2022 at 18:24