# How does adding a primitive root of unity to a number field effect the ring of integers?

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.