What's the intuition behind the fact that we don't always have the equality $O_{\mathbb{Q}(\alpha)} = \mathbb{Z}[\alpha]$ ? As in, from where the ``extra" algebraic numbers which somehow happens to be algebraic integers are coming from ?
Same question for the fact that why we don't always have $O_{KL} \neq O_K \times O_L$ (where $[L:\mathbb{Q}] = [KL:L]$), and why you're expected to have an ``extra" family of algebraic integers in $O_{KL}$ that are not in $O_K \times O_L$ ?
I do know examples of the above (for the first line, $\alpha$ being a root of $x^3+x^2-2x-8 =0$, and for the second case taking $K = \mathbb{Q}[\sqrt{2}], L = \mathbb{Q}[\sqrt{3}]$ works (we get an ``extra" family of algebraic integers $\frac{\sqrt{2}+\sqrt{6}}{2}$ in $O_{KL}$, and that's because we have the identity $\sqrt{2+\sqrt{3}} = \frac{\sqrt{2}+\sqrt{6}}{2}$, which to me is quite surprising and/or coincidental)), but I don't have any intuition as why such things are happening, and whether this event is the exception or the norm.
I would think that $O_{\mathbb{Q}[\alpha]} \neq \mathbb{Z}[\alpha]$ is an exception, since taking the minimal polynomial of any element $z = \sum a_k \alpha^k$ in $K$ over $\mathbb{Q}$ and needing them to be integers AND in order for $z \not \in \mathbb{Z}[\alpha]$ requires $a_k$'s satisfying some very rigid set of equation.