TITLE: Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.
QUESTION [7 upvotes]: I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the galois extension for $X^3-2$). Does anyone have a reference or proof?

REPLY [8 votes]: Let $L$ be the splitting field of $x^3-2$. In this case, $\mathcal{O}_L$ has a power basis. Indeed, $\mathcal{O}_L=\mathbb{Z}[\varepsilon]$ where $\varepsilon$ is a root of 
$$f(x)=x^6 + 3x^5 - 5x^3 + 3x + 1.$$
You can find a proof of this fact, and a whole other bunch of interesting facts about the splitting field of $x^3-2$, in these notes by my colleague, Keith Conrad. (Note that the $K=\mathbb{Q}(\sqrt[3]{2})=\mathbb{Q}(\sqrt[3]{-2})$, so the splitting field of $x^3-2$ and the splitting field of $x^3+2$ coincide.)