Definition:Root of Unity

Definition

Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $F$ be a field.

The $n$th roots of unity of $F$ are defined as:

$U_n = \set {z \in F: z^n = 1}$

Complex Roots of Unity

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

The complex $n$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^n = 1}$

Primitive Root of Unity

A primitive $n$th root of unity of $F$ is an element $\alpha \in U_n$ such that:

$U_n = \set {1, \alpha, \ldots, \alpha^{n - 1} }$

Order of Root of Unity

Let $z \in U_n$.

The order of $z$ is the smallest $p \in \Z_{> 0}$ such that:

$z^p = 1$

Examples

Square Root

The square roots of unity of $F$ are defined as:

$U_2 = \set {z \in F: z^2 = 1}$

Cube Root

The cube roots of unity of $F$ are defined as:

$U_3 = \set {z \in F: z^3 = 1}$

Also see

• Results about roots of unity can be found here.

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.4$ Hensel's Lemma
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): root of unity
• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): root of unity