How to determine whether an extension is normal extension.

Question

Let $\alpha = e^{2 \pi \imath/19} + e^{16 \pi \imath/19} + e^{14 \pi \imath/19} + e^{36 \pi \imath/19} + e^{22 \pi \imath/19} + e^{24 \pi \imath/19}$, its minimal polynomial is $f(x) = x^3 + x^2 - 6x - 7$, so $[\mathbb{Q}(\alpha):\mathbb{Q}] = 3$. How could I determine whether or not is beforementioned extension normal? I know that it would be normal, if it is a splitting field for that minimal polynomial, but since its roots are a bit crazy, I guess there must be simpler way.

Answer 1

The field $K_n=\Bbb Q(e^{2\pi i/n})$ is an extension of $\Bbb Q$ with Abelian Galois group. Any subextension is normal (and Abelian). In your example $n=19$.