Q.Find a polynomial equation of minimum degree with rational coefficients, having $2-\sqrt3$ as a root.

Theorem. If an equation with rational coefficients has an irrational root $a+\sqrt b$ (with $a,b$ rational and $\sqrt b$ irrational), then $a-\sqrt b$ is also a root, with the same multiplicity.

Proof idea. This runs exactly parallel to the complex case. Divide $f(x)$ by the rational quadratic $(x-a)^2-b=x^2-2ax+(a^2-b)$, the minimal rational polynomial having $a+\sqrt b$ as a root:

$f(x)=\big[(x-a)^2-b\big]Q(x)+Rx+S,$ …