Determine the minimal polynomial of $\sqrt 3+\sqrt 5$ I am struggling in finding the minimal polynomial of $\sqrt{3}+\sqrt{5}\in \mathbb C$ over $\mathbb Q$
Any ideas? I tried to consider its square but it did not helped..

I propose the following way to prove the polynomial $\;x^4-16x^2+4\;$ is irreducible over $\;\Bbb Q\;$ . First, factor it over the reals:
$$x^4-16x^2+4=(x^2-2\sqrt5\,x+2)(x^2+2\sqrt5\,x+2)$$
(this is way easier than what can thought at first, at least in this an other similar cases).
From here, it's clear the polynomial cannot be factores any other essentially different way as $\;\Bbb R[x]\;$ is a UFD and, of course, $\;\Bbb Q\subset\Bbb R\;$ . End the argument now.
