The Galois group of the field extension of the rational numbers is isomorphic to the multiplicative group modulo "n", which has order φ ("n ") and is cyclic for some but not all"n".
The Galois group of the field extension of the rational numbers generated by the "n" th roots of unity forms a different group. It is isomorphic to the multiplicative group modulo "n", which has order φ ("n ") and is cyclic for some but not all"n".