Let $\alpha = a + b \sqrt{d} \in \mathbb{Q} \left(\sqrt{d} \right) = \{a+b \sqrt{d}:a,b \in \mathbb{Q} \}.$
The minimal polynomial $m(x)$ of an algebraic number $\alpha \in \mathbb{C}$ is the monic polynomial of smallest degree, with coefficients in $\mathbb{Q}$ such that $m(\alpha) = 0.$ How do we find the minimal polynomial of $\alpha = a + b \sqrt{d}$ over $\mathbb{Q}$?
Best Answer
$$\alpha-a=b\sqrt{d}$$ $$(\alpha-a)^2=b^2d$$ $$\alpha^2-2\alpha a+a^2=b^2d$$ $$f(x)=x^2-2x a+a^2-b^2d$$