Let $$\alpha=(\sqrt[713]{5}+\sqrt{2}+i)\sqrt[13]{100001}.$$ Is this number an algebraic integer over $\mathbb{Z}$?
I have really no clue, because I do not find any polynomial in $\mathbb{Z}[x]$ which has $\alpha$ as a root. Maybe I have to consider the norm, but this does not bring me further.
Of course it is hard to find such a polynomial. But it is well known that the algebraic integers form a ring, they are closed under addition and multiplication. It is not hard to check that the four constants which appear in the definition of $\alpha$ are indeed algebraic integers, so $\alpha$ is an algebraic integer as well.