How can I find the splitting field of $x^n+1 \in \mathbb{Q}[x]$ ??
If we have for example, $x^n-1 \in \mathbb{Q}[x]$ we would do the following:
$$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \dots +x+1)$$
So, the splitting field is $\mathbb{Q}(1, \omega_n, \omega_n^2, \dots, \omega_n^{n-1})=\mathbb{Q}(\omega_n)$, where $\omega_n=e^{\frac{2\pi i}{n}}$.
How can we find the splitting field in the case $x^n+1 \in \mathbb{Q}[x]$ ??
I believe $\Bbb {Q}[e^{\frac{\pi i}{n}}]$ is all you need.
We need to solve $x^n=-1$, but $-1=e^{\pi i}$. So raising each side to the $\frac{1}{n}$ gives $x=e^{\frac{\pi i}{n}}$.