Let $f(x)=(x^6+x^2)(x^5-6x+10)$. Determine all of the maximal ideals of $\mathbb{Q}(i)[x]$ that contain $f(x)$.
My Attempt:
Since, I find that $f(x)=(x^6+x^2)(x^5-6x+10) = x^2(x^4 + 1)(x^5-6x+10) = x^2 (x^2 + i)(x^2 - i)(x^5-6x+10) = x^2 (x+(\frac{\sqrt{2}}{2} \pm (\frac{1}{4})^{1/4}i))(x-(\frac{\sqrt{2}}{2} \pm (\frac{1}{4})^{1/4}i))(x^5-6x+10)$.
So, are that $\langle x \rangle$, $\langle x + (\frac{\sqrt{2}}{2} \pm (\frac{1}{4})^{1/4}i) \rangle$, $\langle x -(\frac{\sqrt{2}}{2} \pm (\frac{1}{4})^{1/4}i) \rangle$, and $\langle x^5-6x+10 \rangle$ the only maximal ideals of $\mathbb{Q}(i)[x]$ that contain $f(x)$?