Theorem 2 (Gauss)

— Let $Q$ be an algebraic closure of $Q$ and let $n \geq 1$ be an integer.

a) The cyclotomic polynomial $\Phi_n(X)$ is irreducible in $Q[X]$.

b) The degree of $R_n(Q)$ over $Q$ is $\varphi(n)$.

c) The homomorphism $\chi_n$ of $\mathrm{Gal}(\bar{Q}/Q)$ into $(\mathbf{Z}/n\mathbf{Z})^*$ is surjective and defines by passage to quotients an isomorphism of $\mathrm{Gal}(R_n(Q)/Q)$ onto $(\mathbf{Z}/n\mathbf{Z})^*$.

Taking account of Prop. 6, we need only prove c). Every integer $r$ prime to $n$ is a product of prime numbers $p_1, \ldots, p_s$ not dividing $n$; so it is enough to show that for every prime number $p$ not dividing $n$, the mapping $x \mapsto x^p$ of $\mu_n(Q)$ into itself extends to an automorphism of $R_n(Q)$. It suffices to prove that if $\zeta$ is a primitive $n$-th root of unity, the minimal polynomial $f$ of $\zeta$ over $Q$ is equal to the minimal polynomial $g$ of $\zeta^p$ over $Q$.

We shall assume that $f \neq g$ and argue by contradiction. The polynomials $f$ and $g$ are monic irreducible in $Q[X]$ and divide $X^n - 1$, so there exists $u \in Q[X]$ such that $X^n - 1 = fg u$ (IV, p. 13, Prop. 13). Lemma 2 (V, p. 82) shows that $f, g$ and $u$ have integer coefficients. Let us denote by $\bar{v}$ the polynomial with coefficients in $\mathbf{F}_p$ obtained from a polynomial $v \in \mathbf{Z}[X]$ by reduction mod $p$. We thus have $X^n - 1 = \bar{f} \bar{g} \bar{u}$ in $\mathbf{F}_p[X]$.

Further we have $g(\zeta^p) = 0$ and so $g(X^p)$ is a multiple of $f(X)$ in $\mathbf{Q}[X]$. By Lemma 2 there exists $h \in \mathbf{Z}[X]$ such that $g(X^p) = f(X) \cdot h(X)$. Now we have $v(X^p) = v(X)^p$ for every polynomial $v \in \mathbf{F}_p[X]$. By reduction mod $p$ we thus obtain $\bar{g}^p = \bar{f} \bar{h}$. If $v$ is an irreducible polynomial in $\mathbf{F}_p[X]$ dividing $\bar{f}$, it must then divide $\bar{g}$. Since $\bar{f} \bar{g}$ divides $X^n - 1$, we conclude that $v^2$ divides $X^n - 1$ in $\mathbf{F}_p[X]$. This is absurd because the polynomial $X^n - 1$ is separable in $\mathbf{F}_p[X]$.

It may be shown that for every abelian extension $E$ of finite degree over $Q$ there exists an integer $n \geq 1$ such that $E$ is isomorphic to a subextension of $R_n(Q)$. \* In other words, the field $Q(\mu_{\infty}(C))$ is an abelian closure of $Q$. *(« Kronecker-Weber theorem »).