power basis over

Let $K$ be a number field with $[K\!:\!\mathbb{Q}]=n$ and $\mathcal{O}_K$ denote the ring of integers of $K$ . Then $\mathcal{O}_K$ has a power basis over $\mathbb{Z}$ (sometimes shortened simply to power basis) if there exists $\alpha \in K$ such that the set $\{ 1, \alpha, \ldots, \alpha^{n-1}\}$ is an integral basis for $\mathcal{O}_K$ . An equivalent condition is that $\mathcal{O}_K=\mathbb{Z}[\alpha]$ . Note that if such an $\alpha$ exists, then $\alpha \in \mathcal{O}_K$ and $K=\mathbb{Q}(\alpha)$ .

Not all rings of integers have power bases. (See the entry biquadratic field for more details.) On the other hand, any ring of integers of a quadratic field has a power basis over $\mathbb{Z}$ , as does any ring of integers of a cyclotomic field. (See the entry examples of ring of integers of a number field for more details.)

power basis over is owned by Warren Buck.