Corollary 2

For every integer $n \geq 1$, we have

where the integer $d$ runs over the set of divisors > 0 of $n$ ^1.

With the notation of Cor. 1, every element of $G$ has a finite order which is a divisor $d$ of $n$ and for a fixed $d$ there are $\varphi(d)$ such elements.

The calculation of $\varphi(n)$ is based on the two formulae:

(2) $\varphi(mn) = \varphi(m)\varphi(n)$ if $m$ and $n$ are relatively prime , (3) $\varphi(p^a) = p^{a-1}(p-1)$ ($p$ prime, $a \geq 1$).

The first follows at once from the fact that the rings $\mathbf{Z}/mn\mathbf{Z}$ and $(\mathbf{Z}/m\mathbf{Z}) \times (\mathbf{Z}/n\mathbf{Z})$ are isomorphic (I, p. 112), and that $(A \times B)^* = A^* \times B^*$ for two rings $A$ and $B$. To prove (3) we note that the positive divisors of $p^a$ are $1, p, p^2, \ldots, p^a$; hence the integer $k$ has no common divisor with $p^a$ other than $1$ if and only if it is not divisible by $p$; since there are $p^a - 1$ multiples of $p$ between $0$ and $p^a - 1$, we find indeed (3).

The formulae (2) and (3) show at once that

where $p$ runs over the set of prime divisors of $n$.

^1 The relation $d \mid n$ between integers > 0 means « d divides n » (VI, p. 5).

An n-th root of unity is said to be primitive if it is of order $n$; if there exists such a root $\zeta$, the group $\mu_n(K)$ is of order $n$ and is generated by $\zeta$. \* For example, the primitive n-th roots of unity in $\mathbf{C}$ are the numbers $e^{2\pi i k / n}$ with $0 \leq k < n$ and $k$ prime to $n$. \* Cor. 1 of Prop. 3 now implies the following result.