I was tasked with finding $\min(\mathbb{Q},i+\sqrt[]{2})$

I found the polynomial in $\mathbb{Z}[X]$ to be $X^4-2X^2+9$

I know it must be minimal over $\mathbb{Q}$ since (used brute force) any arbitrary qubic, quadratic, or linear polynomial has coefficients in $\mathbb{C}-\mathbb{Q}$

My question is is there an easier method?

(Edit: question initially asked about irreducibility, but since minimal implies irreducibility, we make the change)