Trying to find splitting fields over Q of $x^{19} -1$

Question

I'm trying to find subfields L of C which are splitting fields over Q

For $x^{19}-1$

I've found the roots, but since you can't express them in exact form I don't see what to do next.

Answer 1

First of all, the splitting field is unique, and it is the unique extension of $\mathbb{Q}$ containing the roots of the polynomial $x^{19}-1$.

Now, if you call $$\omega = e^{\frac{2 \pi i}{19}} = \cos\frac{2}{19}\pi + i \sin \frac{2}{19}\pi$$ then $\omega^{19} = e^{2 \pi i} = 1$, i.e. $\omega$ is a root of $x^{19}-1$.

With the same argument you can prove that $\{ \omega , \omega^2 , \dots , \omega^{18}, \omega^{19}=1\}$ are all the distinct roots of your polynomial. So your splitting field is $L= \mathbb{Q}(\omega , \omega^2 , \dots) = \mathbb{Q}(\omega )$.