# Integral Basis for Cubic Fields

I'm trying to follow a text (Lang's Algebraic Number Theory) in which it fully determines an integral basis for quadratic fields (also seen here). Is there any easy or analogous way to determine one for cubic fields of the form $\mathbb Q(\sqrt[3]{a})$, where $a\in\mathbb Z$?

Can one also conclude (or stipulate various restrictions so) that $\mathcal O_K$ is a PID?

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math.ku.dk/~kiming/lecture_notes/… – user27126 Dec 4 '12 at 3:09
$\mathcal{O}_K$ is not in general a PID when $K$ is quadratic, so there should be less hope for when $K$ is cubic. – Rankeya Dec 4 '12 at 4:06
If you know about the ideal class group, then $\mathcal{O}_K$ is a PID if and only if its class number (i.e. the order of the ideal class group) is 1. This wikipedia article has a list of some quadratic and cubic fields with class number 1: en.wikipedia.org/wiki/… – Rankeya Dec 4 '12 at 4:09

Integral basis for ${\bf Q}(\root3\of a)$ is given in Theorem 7.3.2 of Alaca and Williams, Introductory Algebraic Number Theory:
Let $m$ be a cubefree integer. Set $m=hk^2$, where $h$ is squarefree, so that $k$ is squarefree and $(h,k)=1$. Set $\theta=m^{1/3}$ and $K={\bf Q}(\theta)$. Then an integral basis for $K$ is \eqalign{&\{{1,\theta,\theta^2/k\}},{\rm\ if\ }m^2\not\equiv1\pmod9,\cr&\{{1,\theta,(k^2\pm k^2\theta+\theta^2)/3k\}},{\rm\ if\ }m\equiv\pm1\pmod9.\cr}