## Construction of Polynomial With Root sqrt 3 + sqrt 2 Which Has Integer Coefficients

Is it possible to construct a polynomial with a root
$\sqrt{3} + \sqrt{2}$
(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{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}

This is the required polynomial.