Is $\sqrt{2+\sqrt{2}}+\sqrt[3]{3}/2$ an algebraic integer?

Question

I know that $\sqrt{2+\sqrt{2}}$ is, and that $\sqrt[3]{3}/2$ is not, but what can I say about the sum of the two?

Answer 1

$\sqrt{2+\sqrt{2}}+\frac{ \sqrt[3]{3}}{2}$ is an algebraic integer iff

$\frac{\sqrt[3]{3}}{2}$ is an algebraic integer (since $\sqrt{2+\sqrt{2}}$ is obviously an algebraic integer).

But the minimal polynomial over $\mathbb{Q}$ of $\frac{ \sqrt[3]{3}}{2}$ is: $$ 2x^3-3; $$ So it is not algebraic integer (you have the $2$ as leading coefficient, you can not simplify it since $\text{GCD}(2,3)=1$ ).