Show that $f(t)=t^8-2t^4+9$ is the minimal polynomial of $\alpha = \sqrt{i+\sqrt 2}$ over $\mathbb{Q}$

Question

I am really struggling to show that. I can't find a proof for $f$ to be irreducible. Eisentsien doesn't work. Revesing doesn't lead me anywhere and mod p didn't work as well, is there any criterion I might be missing? I have all the roots of $f$, but I don't think that is very useful for thir step.

Answer 1

There is a general naive algorithm for testing the irreducibility of integer polynomials (which also works over number fields).

If $f$ is not irreducible then there is $g\in \Bbb{Z}[t]$ monic of degree $\le 4$ dividing it and whose roots have absolute value $\le 3^{1/4}$.

This implies that the coefficients of $g$ are smaller than those of $(t+3^{1/4})^{\deg(g)}$.

Just try the finitely many such polynomials and see if they divide $f$.