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Apparently this should be a straightforward / standard homework problem, but I'm having trouble figuring it out.

Let $D$ be a square-free integer not divisible by $3$. Let $\theta = \sqrt[3]{D}$, $K = \mathbb{Q}(\theta)$. Let $\mathcal{O}_K$ be the ring of algebraic integers inside $K$. I need to find explicitly elements generating $\mathcal{O}_K$ as a $\mathbb{Z}$-module.

It is reasonably clear that $\theta$ is itself an algebraic integer and that $\mathbb{Z}[\theta] \le \mathcal{O}_K$, but I strongly suspect it isn't the whole ring. I'm not sure where the hypotheses on $D$ come in at all... any hints would be much appreciated.

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    A general element of $\mathbb{Q}(\theta)$ has the form $a+b\theta+c\theta^2$ for some $a,b,c \in \mathbb{Q}$. Work out the minimal polynomial of such an element, and check when it's monic.2011-02-12
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    I don't think there is any really clean way to do this computation. If you look at http://www.math.uconn.edu/~kconrad/blurbs/ , at the files entitled "Invariants of the Splitting Field of a Cubic n", for $1 \leq n \leq 5$, you'll find many worked examples.2011-02-12
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    @David Speyer: Those look nice, but unfortunately I know nothing about local fields or ramification of primes. Yet. Perhaps I'll come back to them when I know more.2011-02-14
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    When $D = 2$, for example, $\mathbf Z[\theta]$ is the whole thing.2012-01-23

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