Let be a number field with
and
denote the ring of integers of . Then
has a power basis over (sometimes shortened simply to power basis) if there exists
such that the set
is an integral basis for
. An equivalent condition is that
. Note that if such an exists, then
and
.
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
![$ [K\!:\!\mathbb{Q}]=n$](http://images.planetmath.org:8080/cache/objects/7960/l2h/img2.png "$ [K\!:\!\mathbb{Q}]=n$"){kind=small}
![$ \mathcal{O}_K=\mathbb{Z}[\alpha]$](http://images.planetmath.org:8080/cache/objects/7960/l2h/img10.png "$ \mathcal{O}_K=\mathbb{Z}[\alpha]$"){kind=small}
