power basis over $\mathbb{Z}$

Let $K$ be a number field with $[K:\mathbb{Q}]=n$ and ${𝒪}_K$ denote the ring of integers of $K$. Then ${𝒪}_K$ has a power basis over $\mathbb{Z}$ (sometimes shortened simply to power basis) if there exists $\alpha \in K$ such that the set $\{1,\alpha ,\mathrm{\dots },{\alpha }^{n-1}\}$ is an integral basis for ${𝒪}_K$. An equivalent (http://planetmath.org/Equivalent3) condition is that ${𝒪}_K=\mathbb{Z}[\alpha ]$. Note that if such an $\alpha$ exists, then $\alpha \in {𝒪}_K$ 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 $\mathbb{Z}$
Canonical name	PowerBasisOvermathbbZ
Date of creation	2013-03-22 15:56:55
Last modified on	2013-03-22 15:56:55
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	17
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 11R04
Synonym	power basis
Synonym	power bases
Related topic	ConditionForPowerBasis