TITLE: Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?
QUESTION [6 upvotes]: Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? 
More generally, for any Salem number $s$, does the ring of integers for $\mathbb{Q}(s)$ have a power basis?
Does anyone know of a sage command or some clever way to determine if any given ring of integers for an algebraic number field has a power basis?
Thanks!

REPLY [5 votes]: Let $u = (\sqrt{-1+2\sqrt 2}+\sqrt 2+1)/2$.
The minimal polynomial of $u$ over $\Bbb Q(\sqrt 2)$ is $u^2-u(1+\sqrt 2)+1$, and over $\Bbb Q$ it is $u^4-2u^3+u^2-2u+1$, whose discriminant is $-448$.
Since this is the discriminant of the number field, its ring of integers is $\Bbb Z[u]$.