We know that if $xi$ is a primitive $n^text{th}$-root of unity, then the ring of integers $mathcal{O}_{mathbb{Q}(xi)}$ of $mathbb{Q}(xi)$ is $mathbb{Z}(xi)$.

Can we generalise this result to say much about the ring of integers $mathcal{O}_{K(xi)}$ of $K(xi)$, where $K / mathbb{Q}$ is some finite algebraic extension?

Is it the case that $mathcal{O}_{K(xi)} = mathcal{O}_{K}(xi)$?

If this is not generally true, do we have a characterisation of circumstances where this may hold?

Failing that, do we have an alternate description of $mathcal{O}_{K(xi)}$ in terms of $mathcal{O}_{K}$?

I would appreciate any comments, or even just a reference for these kinds of results.