Can you find a polynomial (not necessarily a quadratic) with integer coefficients which has \(\sqrt{2}+\sqrt{3}\) as a root? What other roots (if any) does it have?

Can you find a polynomial with integer coefficients which has \(1+\sqrt[3]{2}=1+2^{\frac{1}{3}}\) as a root? What other roots (if any) does it have?

Can you find a polynomial with integer coefficients which has \(\sqrt[3]{2}+\sqrt[3]{4}=2^{\frac{1}{3}}+2^{\frac{2}{3}}\) as a root? What other roots (if any) does it have?

Can you make up some similar questions to the above?

Are there any numbers for which you

*cannot*find a polynomial with integer coefficients which has that number as a root?