Prove that $\mathcal{O}_K$ of a cubic field $\mathbb{Q}(\alpha)$ is $\mathbb{Z}[\alpha]$

Question

Suppose $\alpha \in K= \mathbb{Q}(\alpha)$ satisfies $\alpha^3-9\alpha-6$. The discriminant of $\mathbb{Z}[\alpha]$ is $-1944=2^3\cdot 3^5$ which is not a square-free and in particular Stickelberg's identity doesn't work. From $$\text{dist}(\mathbb{Z}[ \alpha])=\text{dist}(O_K)[\mathcal{O}_K:\mathbb Z[\alpha]]^2$$ I know that $[\mathcal{O}_K:\mathbb Z[\alpha]] \mid 18$. I found several posts answering similar questions using more advanced arguments (e.g. considering prime ideals $\mathfrak{p}$ and ramification index at $\mathfrak{p}$) but honestly I'm not familiar with that methods. Can anyone provide some extra guides for me? Thank you.

Answer 1

Here is an elementary approach.

Start by writing down a general element of $\Bbb Q(\alpha)$, which looks like $x=a+b\alpha+c\alpha^2$ with $a,b,c\in \Bbb Q$.

If $y,z$ are the two conjugates of $x$, then $x$ is integral iff $x+y+z,xy+yz+zx,xyz$ are all integers. You may then compute these three expressions as:

• $x+y+z=3a$;
• $xy+yz+zx=3a^2-9b^2+36ac-18bc+81c^2$;
• $xyz=a^3-9ab^2+6b^3+18a^2c-18abc+81ac^2-54bc^2+36c^3$.

Thus it suffices to show that $a,b,c$ must be integers when the three quantities above are integers.

If you use your knowledge that $18\mathcal O_K\subseteq \Bbb Z[\alpha]$, then we already know that $18a,18b,18c$ are integers, and it becomes a case-by-case verification that the remainders of $18a,18b,18c$ modulo $18$ can only be zeros.

It can be done more efficiently (by considering mod $2$, mod $3$, then mod $9$), but after all it's an elementary yet really complicated exercise, and in the end the main argument should be equivalent to the "more advanced arguments" you mentioned in your post.

Thus I suggest that you learn these more advanced things, which will be useful for other tasks as well.