# Ideal classes in the ring of integers in quadratic number fields

1. Jun 5, 2009

### sidm

So I've been working through some algebraic number theory and there is a problem asking me to describe the ideals in the following rings.

a) The ring of algebraic integers in $Q[\sqrt{-3}]$ call it $R$ or equivalently $Z[\frac{1}{2}(1+\sqrt{-3})]$ where $Z$ is the ring of ordinary integers.

I say they are either the principle ideals generated by an element of minimum modulus $\alpha$. Otherwise it isn't a principle ideal and we can describe it's lattice basis: $(\alpha, \frac{1}{2}\alpha(1+\sqrt{-3})$. These are the only two possiblities, am i correct? I could reproduce the argument but it relies too much on machinery developed in Artin's Algebra text (specifically chapter 11) to be self-contained here.

b) Here we are looking at the ideals in the integers adjoin the root of negative six. That is the ideals in $Z[\sqrt{-6}]$. Let $I$ and $\alpha$ be as in (a) then either the ideal is principle generated by $\alpha$ or it isn't and there are two possiblities:

the lattice basis for $I$ is $(\alpha, \frac{1}{2}\alpha\delta)$ where $\delta =\sqrt{-6}$. Otherwise it is $(\alpha, \frac{1}{2}(\alpha+\delta\alpha))$. In other words the class number here is three.

This is difficult to get feedback on as I'm working on my own. So any help would be appreciated.