What is the splitting field of the polynomial $X^2-X+3 \in \mathbb{Q}[X]$ equal to?

Question

What is the splitting field of the polynomial $X^2-X+3 \in \mathbb{Q}[X]$ equal to?

What I have tried:

I factorised the polynomial to get $X= \pm \frac{i\sqrt{11}}{2}$. Does this mean the answer is $\mathbb{Q}[i\sqrt{11}]$?

Answer 1

Yes, it does. $\mathbb{Q}[i\sqrt{11}]$ is the splitting field of your polynomial $p(x)$ over $\mathbb{Q}$ since it is the smallest field containing $\mathbb{Q}$ and the roots of $p(x)$. As $\mathbb{Q}[i\sqrt{11}]$ contains $\mathbb{Q}$, it contains $\frac{1}{2}$ and, because it is a field it contains $\frac{1\pm i \sqrt{11}}{2}$.