Find inverse with minimal polynomial

Question

Give $a=\sqrt{3}+\sqrt{5}$, I need to find the minimal polynomial of $a$ in $\mathbb{Q}$ and the polynomial $f\in\mathbb{Q}[X]$ with $f(a)=a^{-1}$.

Now the first part is rather easy, but how do I calculate the second part?

Answer 1

Give $a=\sqrt3+\sqrt5$, I need to find the minimal polynomial of $a$.

$$\begin{align} a=\sqrt3+\sqrt5 &\quad\Rightarrow\quad a^2=(\sqrt3+\sqrt5)^2 = 8+2\sqrt{15}\\ &\quad\Rightarrow\quad (a^2-8)^2 = 4\cdot15\\ &\quad\Rightarrow\quad 0=a^4-16a^2+4\\ \end{align}$$

$a^{-1}$

$$0=a^4-16a^2+4 \;\Leftrightarrow\; 0=a^3-16a+4a^{-1}$$ Thus $$a^{-1}=-\frac14a^3+4a$$