Find the minimal polynomial of the 18th root of unity and the Galois group of $\mathbb{Q}(z_{18})$
So the minimal polynomial is $x^6-x^3+1$ and i checked online and it is correct. Now for the galois group it is isomorphic to $(Z_{18})^*\approx(Z_9)^*\approx Z_6$. Thus it is cyclic. Its generators are $z_{18}\to z_{18}^5$ and $z_{18}\to z_{18}^{11}$
