Definition 2

An element $\zeta$ of K is said to be a root of unity if there exists an integer $n > 0$ such that $\zeta^n = 1$; for every integer $n > 0$ such that $\zeta^n = 1$, $\zeta$ is called an n-th root of unity.

It amounts to the same to say that the roots of unity are the elements of finite order of the multiplicative group $K^*$ of non-zero elements of K (I, p. 51). The roots of unity form a subgroup $\mu_\infty(K)$ of $K^*$, the n-th roots form a subgroup $\mu_n(K)$ of $\mu_\infty(K)$. We have $\mu_\infty(K) = \bigcup_{n \geq 1} \mu_n(K)$ and $\mu_n(K) \subset \mu_m(K)$ if $n$ divides $m$. For every root of unity $\zeta$ there exists a least integer $n \geq 1$ such that $\zeta$ belongs to $\mu_n(K)$, namely the order of $\zeta$ in the group $K^*$.

The group $\mu_n(K)$, being the set of roots of the polynomial $X^n - 1$, is of finite order $\leq n$. Let $p$ be the characteristic of K. When $p = 0$, or when $p \neq 0$ and $n$ is not divisible by $p$, the derivative $nX^{n-1}$ of $X^n - 1$ is relatively prime to $X^n - 1$ and so the polynomial $X^n - 1$ is then separable; if moreover K is algebraically closed, $\mu_n(K)$ is thus a group of $n$ elements.

Suppose that K is of non-zero characteristic $p$ and let $r \geq 0$ be an integer; since the mapping $x \mapsto x^{p^r}$ of K into K is injective, we have $\mu_{np^r}(K) = \mu_n(K)$ for every integer $n \geq 1$.

We remark that a field may contain no n-th root of unity other than 1: this is the case for example with the prime fields $\mathbf{Q}$ and $\mathbf{F}_p$ for any odd integer $n$.