Construction of Polynomial With Root sqrt 3 + sqrt 2 Which Has Integer Coefficients
\[\sqrt{3} + \sqrt{2}\]
(among others) which has integer coefficients?Yes it is.
Let two of the roots be
\[\alpha = \sqrt{3} + \sqrt{2}, \; \beta = \sqrt{3} - \sqrt{2} \]
.\[(x- \alpha )(x- \beta )=x^2-( \alpha + \beta ) x + \alpha \beta= x^2-2x \sqrt{3}+1\]
This polynomial does not have integer coefficients, but consider
\[\begin{equation} \begin{aligned} (x^2-2x \sqrt{3}+1)(x^2+2x \sqrt{3}+1) &= (x^2+1-2x \sqrt{3})(x^2+1+2x \sqrt{3}) \\ &= (x^2+1)^2-(2x \sqrt{3})^2 \\ &= x^4+2x^2+1-12x^2 \\ &= x^4-10x^2+1 \end{aligned} \end{equation}\]
This is the required polynomial.
