The group ${\mu }_n(K)$ of $n$-th roots of unity in a field, and primitive $n$-th roots of unity

Let $K$ be a field and let $n\ge 1$ be an integer. An element $x\in K$ is an $n$-th root of unity when $x^n=1$, that is when $x$ is a root of $t^n-1\in K[t]$ (Evaluation and roots of a polynomial in a commutative target ring). Write

${\mu }_n(K):=\{x\in K:x^n=1\}\mathrm{.}$

This is a subgroup of $K^{\times }$ (Subgroup). Every $x\in {\mu }_n(K)$ is invertible, with inverse $x^{n-1}$ (Left inverse, right inverse, and invertible element of a monoid), so ${\mu }_n(K)\subseteq K^{\times }$; it contains $1$; it is closed under multiplication, since $(xy{)}^n=x^n{y}^n=1$; and it is closed under inverses, since $(x^{-1}{)}^n=(x^n{)}^{-1}=1$.

An element $\zeta \in {\mu }_n(K)\subseteq K^{\times }$ is a primitive $n$-th root of unity when its order in the group $K^{\times }$ is exactly $n$ (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity):

$\mathrm{ord}(\zeta )=n\mathrm{.}$

Equivalently, ${\zeta }^n=1$ and no exponent $k$ with $1\le k<n$ has ${\zeta }^k=1$.

An $n$-th root of unity need not be primitive. In $\mathbb{Q}$ the element $-1$ is a fourth root of unity of order two, not four; and ${\mu }_n(K)$ may consist of $1$ alone, as ${\mu }_3(\mathbb{Q})$ does. The two notions are separated deliberately, and the exact circumstances under which a primitive $n$-th root of unity exists are the content of ${\mu }_n(K)$ is cyclic of order dividing $n$, and has a primitive $n$-th root of unity exactly when its order is $n$ and $t^n-1$ is separable over $K$ exactly when the characteristic does not divide $n$, and then a splitting field carries $n$ distinct $n$-th roots of unity.

• The concrete complex picture arrives later in the reading order. Over $\mathbb{C}$ the $n$-th roots of unity are the numbers $e^{2\pi ik\mathrm{/}n}$, and that description is developed in The $n$-th roots of a complex number and the $n$ distinct roots of unity for every $n\ge 1$ ↗. Nothing on this page uses it: the definition above is purely algebraic and applies to every field, including those of positive characteristic, where the count of $n$-th roots of unity can be smaller than $n$.