Galois theory and cyclic groups

Just an excerpt from Wikipedia... I am currently reading the book "Symmetry: A Mathematical Exploration" (Springer, Kristopher Tapp)". Is a good reading for beginners (like myself). Amazingly thanks to that great book I started to understand this kind of relationships:

"An $n^{th}$ root of unity may be thought of as a complex number whose $n^{th}$ power is $1$. That is, it is a root of the polynomial $x^n − 1$. The $n^{th}$ roots of unity form a cyclic group of order $n$ under multiplication. For example, the polynomial $0 = z^3 − 1$ factors as $(z − s^0)(z − s^1)(z − s^2)$, where $s = e^{2 \pi i / 3}$; the set $\{ s^0, s^1, s^2 \}$ forms a cyclic group under multiplication. 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 $\phi(n)$ and is cyclic for some but not all $n$."