TITLE: What other kinds of cubic integer rings are there?
QUESTION [7 upvotes]: Given an integer $n \in \Bbb{Z}$, we understand $\root 3 \of n$ to mean the number $x \in \Bbb{R}$ such that $x^3 - n = 0$. Then $\Bbb{Q}(\root 3 \of n) \subset \Bbb{R}$, right? The same then goes for the ring of algebraic integers $\mathcal{O}_{\Bbb{Q}(\root 3 \of n)}$.
Since $\root 3 \of {-n} = -\root 3 \of n$, it follows that $\Bbb{Q}(\root 3 \of n) = \Bbb{Q}(\root 3 \of {-n})$ and likewise $\mathcal{O}_{\Bbb{Q}(\root 3 \of n)} = \mathcal{O}_{\Bbb{Q}(\root 3 \of {-n})}$. So in order for a ring adjoining a cubic root to $\Bbb{Q}$ to have complex numbers we need a cubic root that is imaginary or complex.
Therefore, given $$\omega = \frac{-1 + \sqrt{-3}}{2}$$ and $n > 1$, the ring of integers of $\Bbb{Q}(\omega \root 3 \of n)$ should contain complex numbers. What about the ring of $\Bbb{Q}(i \root 3 \of n)$? That is a distinct kind of rings than the other two I've mentioned, right?
Is this all correct? What kinds of cubic rings have I overlooked?

REPLY [7 votes]: I am copying the information below from Daniel Harrer's diploma thesis, section 2.3; I believe it is originally due to Delone and Fadeev.
For any integers $a$, $b$, $c$, $d$, there is a commutative associative ring with basis $1$, $\alpha$, $\beta$ and multiplication table
$$\begin{array}{rccc}
\alpha^2 &=& b \alpha &- a \beta &-ac \\
\beta^2 &=& d \alpha &- c \beta &-bd \\
\alpha \beta &=& & & -ad \\
\end{array}$$
All cubic integral domains, and more generally all nilpotent-free cubic rings, are of this form.
These formulas define an integral domain if and only if the cubic polynomial $a z^3+b z^2 y + c z + d$ is irreducible. Indeed, if $\omega$ is a root of the irreducible $a z^3+b z^2 y + c z + d$, then we can embed this ring in $\mathbb{C}$ by $\alpha = - a \omega$ and $\beta = d/\omega$. (This paragraph isn't in Harrer.)
Two such cubic rings are isomorphic if and only if the cubics $ax^3+bx^2y+cxy^2+dy^3$ and $a'x^3+b'x^2y+c'xy^2+d'y^3$ are related by $GL_2(\mathbb{Z})$ change of variables in $(x,y)$. (Here $GL_2(\mathbb{Z})$ is $2 \times 2$ integer matrices with determinant $\pm 1$.) See Harrer for details.
I haven't seen anyone do this, but it seems natural to me: Let $\omega_1$, $\omega_2$, $\omega_3$ be the roots of the cubic $az^3+bz^2+cz+d$ in $\mathbb{C}$. Then $GL_2(\mathbb{Z})$ acts on these by Mobius transformations. If we restrict ourself to cubic fields with one real embedding, then one of the $\omega$'s will be in the upper half plane. We can use the $GL_2(\mathbb{Z})$ action to move it into the fundamental domain $|\mathrm{Re}(\omega)|<1/2$, $|\omega|>1$. So it seems to me that cubic rings with one real embedding  have a natural canonical form where we require $az^3+bz^2+cz+d$ to have a root in the fundamental domain.