Proposition 5

— For every integer $n \geq 1$ prime to $p$ the extension $R_n(K)$ of K is abelian of finite degree, its Galois group is isomorphic to a subgroup of $(\mathbf{Z}/n\mathbf{Z})^*$ and its degree divides the order $\varphi(n)$ of $(\mathbf{Z}/n\mathbf{Z})^*$.

Let Q be an algebraic closure of the field Q of rational numbers, and let n be an integer. Then the cyclotomic polynomial $\Phi_n$ of level $n$ is defined by

where $S_n$ is the set of primitive n-th roots of unity in $Q$. The polynomial $\Phi_n$ is of degree $\varphi(n)$ (Prop. 4). It is clear that $\Phi_n(X)$ is invariant under all automorphisms of $\bar{\mathbf{Q}}$, and so belongs to $\mathbf{Q}[X]$. Since every element $\zeta$ of $S_n$ is a root of the polynomial $X^n - 1$, the polynomial $\Phi_n(X)$ divides $X^n - 1$, and the following lemma shows that $\Phi_n(X)$ is a monic polynomial with integer coefficients.