# power basis over $\mathbb{Z}$

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 (http://planetmath.org/Equivalent3) 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.)

Title power basis over $\mathbb{Z}$ PowerBasisOvermathbbZ 2013-03-22 15:56:55 2013-03-22 15:56:55 Wkbj79 (1863) Wkbj79 (1863) 17 Wkbj79 (1863) Definition msc 11R04 power basis power bases ConditionForPowerBasis