I am trying to find an integral closure of $\mathbb{Z}$ in $\mathbb{Q}[I]$
Wikipedia says that integral closure of $\mathbb{Z}$ in $\mathbb{Q}$ is a ring of integers $O_{\mathbb{Q}}$ and I think that the answer should be $O_{\mathbb{Q}}[I]$
But here it's said that the answer is Z[i], because Z is a UFD, and UFD is closed in its field of fractions.
Here the answer is finitely generated as well.
I see, that $x^n-2=0$ provides $\sqrt[n]{2}$ which lie in algebraic closure of $\mathbb{Z}$. I am afraid that I have messed up with terminology, so could you explain me what's wrong?