Is the extension $\mathbb{Q}(\alpha)/\mathbb{Q}$, where $\alpha = 2\pi i /3$, a splitting extension?

Question

Is the extension $\mathbb{Q(\alpha)}:\mathbb{Q}$ where $\alpha=e^{{2\pi i}/3}$ splitting extension because it has degree 2?

Answer 1

I assume that by "splitting extension" you mean "normal extension", as in "the splitting field of some polynomial". As the extension $\Bbb Q(\alpha)/\Bbb Q$ has degree $2$, it must necessarily be the splitting field of the minimal polynomial of $\alpha$ (which is $x^2 + x + 1$). So the answer is yes.