Minimal polynomial of $\sqrt3 + i$ over $\mathbb{R}$ and $\mathbb{C}$ I was able to show that $x^4-4x^2+16$ is the minimal polynomial of $\sqrt3 + i$ over $\mathbb{Q}$.
Does this imply that this is also the mininal polynomial over $\mathbb{R}$ and $\mathbb{C}$?

No. Hint: Consider $(x-\sqrt{3}-i)(x-\sqrt{3}+i)$ over $\mathbb{R}$. For over $\mathbb{C}$, it will have smaller degree. 
